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Point of intersection of a line and a plane - class-XII

Description: point of intersection of a line and a plane
Number of Questions: 42
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Tags: three dimensional geometry maths applications of vector algebra three dimensional geometry - ii
Attempted 0/42 Correct 0 Score 0

Statement-I: The point $A(3,1,6)$ is the mirror image of the point $B(1,3,4)$ in the plane $x-y+z=5$.
Statement-2: The plane $x-y+z=5$ bisects the line segment joining $A(3,1,6)$ and $B(1,3,4)$.

  1. (1 ) StatementI is true. Statement-1 is true: Statement--2 is a correct explanation for Statement-1.

  2. (2) StatementI is true, Statement-2 is true: Statement-9 is not a correct explanation for statement-1.

  3. (3) Statement--I is true, Statement-2 is false.

  4. (4) StatementI is false. Statement-2 is true.


Correct Option: A
Explanation:

Mid-point of AB=$\begin{array}{l} = \left( {\dfrac{{3 + 1}}{2},\dfrac{{1 + 3}}{2},\dfrac{{4 + 6}}{2}} \right)\ = (2,2,5)\end{array}$
lies on the plane as it satisfies the equation of the plane
and DR s of AB $ = (2, - 2,2)$
DR s of normal to the plane $= (1, - 1,1)$
AB is the perpendicular bisector.
Hence, A is the image of 2 
 

If the points $(1,2,3)$ and $(2,-1,0)$ lie on the opposite sides of the plane $2x+3y-2z=k$, then

  1. $k< 1$

  2. $k> 2$

  3. $k< 1$ or $k> 2$

  4. $1< k< 2$


Correct Option: D
Explanation:

 Given plane equation is $2{x}+3{y}-2{z}-k=0$

$(1,2,3)$ and $(2,-1,0)$ lies on the opposite sides of the plane
$(2(1)+3(2)-2(3)-k)(2(2)+3(-1)-2(0)-k)<0$
$(2-k)(1-k)<0\implies (k-1)(k-2)<0$
$\implies 1<k<2$

If the planes $x - cy - bz = 0,cx - y + az = 0\,$ and $bx + ay - z = 0$ pass through a stright line,then the value of ${a^2} + {b^2} + {c^2} + 2abc\,$ is:

  1. $1$

  2. $2$

  3. $3$

  4. none of these


Correct Option: A
Explanation:
If the planes $(x - cy - bz = 0), (cx - y + az = 0)$ and $(bx + ay - z = 0)$ are in same line.
$\therefore$ They must be collinear.
$\begin{vmatrix}1 & -c & -b\\ c & -1 & a\\ b & a & -1\end{vmatrix} = 0$
$\Rightarrow 1(1 - a^2) + c(-c - ab) -b(ac + b) = 0$
$\Rightarrow 1 - a^2 - c^2 - abc - abc - b^2 = 0$
$\therefore a^2 + b^2 + c^2 + 2abc = 1$
Option A is correct

The point where the line through $A=(3, -2, 7)$ and $B= (13, 3, -8)$ meets the xy-plane

  1. $(\cfrac { 23 }{ 3 } ,\cfrac { 1 }{ 3 } ,0)$

  2. $(\cfrac { 23 }{ 6 } ,\cfrac { 1 }{ 6 } ,0)$

  3. $(\cfrac { 23 }{ 3 } ,\cfrac { 1 }{ 3 } , 1)$

  4. $(\cfrac { 23 }{ 3 } ,\cfrac { 1 }{ 3 } , 3)$


Correct Option: A
Explanation:

Equation of line through $A(3,-2,7)$ and $B(13,3,-8)$ is:

$\cfrac { x-3 }{ 10 } =\cfrac { y+2 }{ 5 } =\cfrac { z-7 }{ -15 }$
When the line meets $x-y$ plane $\Rightarrow z=0$
$\therefore \cfrac { x-3 }{ 10 } =\cfrac { y+2 }{ 5 } =\cfrac { 7 }{ 15 } \quad \quad \Rightarrow x=\cfrac { 23 }{ 3 } ,y=\cfrac { 1 }{ 3 } \quad \quad \Rightarrow (x,y,z)=(\cfrac { 23 }{ 3 } ,\cfrac { 1 }{ 3 } ,0)$

The ratio in which the plane $4x+5y-3z=8$ divides the line joining the points $(-2,1,5)$ and $(3,3,2)$ is

  1. $2 : 1$

  2. $1 : 2$

  3. $-2 : 1$

  4. $3 : 2$


Correct Option: A
Explanation:

We know that the ratio in which the plane $ax+by+cz+d=0$ divides the line segment joining (${x _1},{y _1},{z _1}$) and (${x _2},{y _2},{z _2}$) is

$\begin{array}{l} \dfrac { { -\left( { a{ x _{ 1 } }+b{ y _{ 1 } }+c{ z _{ 1 } }+d } \right)  } }{ { a{ x _{ 2 } }+b{ y _{ 2 } }+c{ z _{ 2 } }+d } }  \ a=4;b=5;c=-3;d=-8;{ x _{ 1 } }=-2;{ y _{ 1 } }=1;{ z _{ 1 } }=5;{ x _{ 2 } }=3;{ y _{ 2 } }=3;{ z _{ 2 } }=2 \ so,\, the\, required\, ratio=\dfrac { { -\left( { 4\left( { -2 } \right) +5\left( 1 \right) -3\left( 5 \right) -8 } \right)  } }{ { 4\left( 3 \right) +5\left( 3 \right) -3\left( 2 \right) -8 } }  \ =\dfrac { { -\left( { -8+5-15-8 } \right)  } }{ { 12+15-6-8 } }  \ =\dfrac { { 26 } }{ { 13 } }  \ =\dfrac { 2 }{ 1 } \ or\ 2:1 \end{array}$

Let the equations of a line and a plane be $\dfrac {x+3}{2}=\dfrac {y-4}{3}=\dfrac {z+5}{2}$ and $4x-2y-z=1$, respectively, then

  1. the line is parallel to the plane.

  2. the line is perpendicular to the plane.

  3. the line lies in the plane.

  4. none of these.


Correct Option: A
Explanation:

Direction ratios of the line is $2i+3j+2k$
and normal of plane is along $4i-2j-k$
Now, $(2i+3j+2k).(4i-2j-k)=8-6-2=0$
Therefore, line is parallel to plane

Ans: A

The ratio in which the plane $r.\left( \hat { i } -2\hat { j } +2\hat { k }  \right) =17$ divides the line joining the points $-2\hat { i } +4\hat { j } +7\hat { k } $ and $3\hat { i } -5\hat { j } +8\hat { k } $ is:

  1. $3:5$

  2. $1:10$

  3. $3:10$

  4. $1:5$


Correct Option: C
Explanation:

Let the plane $r.(i-2j+3k)=17$ divide the line joining the points. 

$-2i+4j+7k$ and $2i-5j+8k$ in the ratio $t:1$ at the point $P.$

$\therefore P$ is $\displaystyle \dfrac { 3t-1 }{ t+1 } i+\dfrac { -5t+4 }{ t+1 } j+\dfrac { 8t+7 }{ t+1 } k.$

This lies on the given plane, 

$\displaystyle \therefore \dfrac { 3t-2 }{ t+1 } .1+\dfrac { -5t+4 }{ t+1 } \left( 2 \right) +\dfrac { 8t+7 }{ t+1 } \left( 3 \right) =17$

$\Rightarrow 3t-2+10t-8+24t+21=17t+17$

$\displaystyle \therefore 20t=17-21+10=6\Rightarrow =\dfrac { 6 }{ 20 } =\dfrac { 3 }{ 10 } $

$\therefore$ required ratio is $3:10$.

Line $\vec r=\vec a+\lambda \vec b$ will not meet the plane $\vec r\cdot \vec n=q$, if-

  1. $\vec b\cdot \vec n=0, \vec a\cdot \vec n=q$

  2. $\vec b\cdot \vec n\neq 0, \vec a\cdot \vec n\neq q$

  3. $\vec b\cdot \vec n=0, \vec a\cdot \vec n\neq q$

  4. $\vec b\cdot \vec n\neq 0, \vec a\cdot \vec n=q$


Correct Option: C
Explanation:

Given line is $\overrightarrow { r } =\overrightarrow { a } +\lambda \overrightarrow { b } $

Substitute it in plane equation $\overrightarrow { r } .\overrightarrow { n } =q$
We get $(\overrightarrow { a } +\lambda \overrightarrow { b } ).\overrightarrow { n } =q$
$\Rightarrow \overrightarrow { a } .\overrightarrow { n } +\lambda (\overrightarrow { b } .\overrightarrow { n } )=q$
If $\overrightarrow { b } .\overrightarrow { n } =0$ and $\overrightarrow { a } .\overrightarrow { n } \neq q$ then the line will not meet the plane
Therefore the correct option is $C$

The ratio in which the plane $\vec r\cdot (\vec i-2\vec j+3\vec k)=17$ divides the line joining the points $-2\vec i+4\vec j+7\vec k$ and $3\vec i-5\vec j+8\vec k$ is-

  1. $1:5$

  2. $1:10$

  3. $3:5$

  4. $3:10$


Correct Option: D
Explanation:

Equation of plane in cartesian form is, $x-2y+3z-17=0$.....$(1)$
Assume this plane $(1)$ divide the line segment joining the points $(-2,4,7)$ and $(3,-5,8)$ in $m:n$ ratio
Therefore,  $\dfrac{m}{n} = \dfrac{-2-2(4)+3(7)-17}{3-2(-5)+3(8)-17}= \dfrac{-3}{10} < 0$
Hence, plane (1) divides the given line segment externally $3:10$ 

The plane $\vec r\cdot \vec n=q$ will contain the line $\vec r=\vec a+\lambda \vec b$, if-

  1. $\vec b\cdot \vec n\neq 0, \vec a\cdot \vec n\neq q$

  2. $\vec b\cdot \vec n=0, \vec a\cdot \vec n\neq q$

  3. $\vec b\cdot \vec n=0, \vec a\cdot \vec n=q$

  4. $\vec b\cdot \vec n\neq 0, \vec a\cdot \vec n=q$


Correct Option: C
Explanation:

Normal of the plane $\vec { r } \cdot \vec { n } =q$ is $\vec { n } $ and direction ratio of the line $\vec { r } =\vec { a } +\lambda \vec { b } $ is $\vec { b } $
Since, line lies in the plane, normal and direction ratios should be perpendicular.
Therefore, $\vec { b } \cdot \vec { n } =0$
Also position vector $\vec { a } $ should lie on plane $\vec { r } \cdot \vec { n } =q$
Therefore, $\vec { a } \cdot \vec { n } =q$

Ans: C

The ratio in which the line segment joining the points whose position vectors are $2\hat i-4\hat j-7\hat k$ and $-3\hat i+5\hat j-8\hat k$ is divided by the plane whose equation is $\hat r\cdot (\hat i-2\hat j+3\hat k)=13$ is-

  1. $13:12$ internally

  2. $12:25$ externally

  3. $13:25$ internally

  4. $37:25$ internally


Correct Option: B
Explanation:

Equation of plane in cartesian form is, $x-2y+3z-13=0$ .....(1)$
Assume this plane (1) divide the line segment joining the points
$(2,-4,-7)$ and $(-3,5,-8)$ in $m:n$ ratio
Therefore,  $\dfrac{m}{n} = \dfrac{2-2(-4)+3(-7)-13}{-3-2(5)+3(-8)-13}= \dfrac{-12}{25} < 0$
Henc,e plane (1) divides the given line segment externally  $12:25$.

Which of the following lines lie on the plane $x+2y-z=0$?

  1. $x-1=y-1=1$

  2. $x-y+z=2x+y-z=0$

  3. $\vec r=2\hat i-\hat j+4\hat k+\lambda (3\hat i+\hat j+5\hat k)$

  4. None of these.


Correct Option: D
Explanation:

consider $P: x+2y-z=0$
direction ratio of normal of $P$ is $(1,2,-1)$

a) for $L _{1}: \dfrac {x-1}{1}=\dfrac {y}{-1}=\dfrac {z-5}{-1}$
Since, $(1,0,5)$ does not lie in $P$ 
Therefore, $L _{1}$ does not lie in $P$

b) for $L _{2}: x-y+z=2x+y-z=0$
direction ratio of $L _{2}$ is $(i-j+k) \times (2i+j-k)=3(j+k)$
since, $(i+2j-k).(3j+3k)=6-3=3 \neq =0$
Therefore, $L _{2}$ does not lie on $P$

c) for $L _{3}:\vec r=2\hat i-\hat j+4\hat k+\lambda (3\hat i+\hat j+5\hat k)$
Since, $(2,-1,4)$ does not lie in $P$ 
Therefore, $L _{3}$  does not lie in $P$

Ans: D

Find the ratio in which the segment joining $(1, 2, -1)$ and $(4, -5, 2)$ is divided by the plane $2x - 3y + z = 4$

  1. $2 : 1$

  2. $3 : 2$

  3. $3 : 7$

  4. $1 : 2$


Correct Option: A

If the given planes $ax+by+cz+d=0$ and $ax+by+cz+d=0$ be mutually perpendicular, then 

  1. $\dfrac{a}{a}=\dfrac{b}{b}=\dfrac{c}{c}$

  2. $\dfrac{a}{a}+\dfrac{b}{b}+\dfrac{c}{c}=0$

  3. $aa+bb+cc+dd=0$

  4. $aa+bb+cc=0$


Correct Option: D
Explanation:
$ax+by+cz+d=0$ and ${ a }^{ 1 }x+{ b }^{ 1 }y+{ c }^{ 1 }z+{ d }^{ 1 }=0$ are mutually perpendicular then their respective norm also will be perpendicular too.
Hence by the perpendicular condition,
${ aa }^{ 1 }+{ bb }^{ 1 }+{ cc }^{ 1 }=0$
Where, $\left< a,b,c \right> $ are the direction ratio of the normal to the plane $ax+by+cz+d=0$ and $\left< { a }^{ 1 },{ b }^{ 1 },{ c }^{ 1 } \right> $ are the direction ratio of the normal to the plane ${ a }^{ 1 }x+{ b }^{ 1 }y+{ c }^{ 1 }z+{ d }^{ 1 }=0$
Correct option will be $(D)$
(But there's a formating error. In the question both plane equation are same, which contradicts the fact that the planes are perpendicular).

The ratio in which the joint of $(2, 1, 5), (3, 4, 3)$ is divided by the plane $2x + 2y - 2z - 1 = 0$

  1. $5 : 12$

  2. $12 : 5$

  3. $5 : 7$

  4. $7 : 5$


Correct Option: C
Explanation:

Let P point lie on a plane $2x+2y-2z-1=0 $and divide $(2, 1, 5), (3, 4, 3)$ in a ratio of  $\lambda :1$
by section formula
P=($\frac { 3\lambda +2 }{ \lambda +1 } ,\frac { 4\lambda +1 }{ \lambda +1 } ,\frac { 3\lambda +5 }{ \lambda +1 } $)
put co-ordinate of P in plane equation
2$\left(\cfrac { 3\lambda +2 }{ \lambda +1 } +\cfrac { 4\lambda +1 }{ \lambda +1 } -\cfrac { 3\lambda +5 }{ \lambda +1 } \right)$=1
by solving above equation,
we get,
 $\lambda=\dfrac{5}{7}$

A straight line $\overline { r } =\overline { a } +\lambda \overline { b } $ meets the plane $\overline { r } .\overline { n } =0$ at a point $p$. The position vector of $p$ is

  1. $\overline { a } +\left( \cfrac { \overline { a } .\overline { n } }{ \overline { b } .\overline { n } } \right) \overline { b } $

  2. $\overline { a } -(\overline { b } .\overline { n } )\overline { b } $

  3. $\overline { a } -\left( \cfrac { \overline { a } .\overline { n } }{ \overline { b } .\overline { n } } \right) \overline { b } $

  4. $\overline { a } +(\overline { b } .\overline { n } )\overline { b } $


Correct Option: C
Explanation:
$\rightarrow \ $ Intersection of $\vec r=\vec a+\lambda \vec b$ and $\vec r.\vec n=0$ is $(\vec a+\lambda \vec b),\vec n=0$
$\Rightarrow \ \vec a.\vec n+\lambda \vec b.\vec n=0\ \Rightarrow \lambda =-\dfrac {\vec a.\vec n}{\vec b.\vec n}$
Putting $\lambda $ in $\vec r=\vec a+\lambda \vec b$
$\vec p=\vec a-\left (\dfrac {\vec a.\vec n}{\vec b.\vec n}\right)\vec b\ \Rightarrow \ (c)$


The distance of the point $(-1,-5,-10)$ from the point of intersection of the line $\dfrac{x-2}{2}=\dfrac{y+1}{4}=\dfrac{z-2}{12}$ and the plane $x-y+z=5$ is

  1. $2\sqrt{11}$

  2. $\sqrt{126}$

  3. $13$

  4. $14$


Correct Option: C
Explanation:

Given line is, $\dfrac{x-2}{2} = \dfrac{y+1}{4} = \dfrac{z-12}{2} = k$ (say)


So any point on this line is given by, $(2k+2, 4k-1, 12k+2)$

Now line intersects the plane $x-y+z=5$

$\Rightarrow 2k+2-(4k-1)+12k+2=5 \Rightarrow k = 0$

Thus point of intersection is, $(2, -1, 2)$  

Therefore required distance is $= \sqrt{(2+1)^2+ (-1+5)^2 + (2+10)^2} = 13$

Hence, option 'C' is correct.

The point of intersection of the line joining the points $(2,0,2)$ and $(3,-1,3)$ and the plane $x-y+z=1$ is

  1. $(3,2,0)$

  2. $(-1,1,3)$

  3. $(1,1,1)$

  4. $(4,2,-1)$


Correct Option: A

The expression in the vector form for the point  $\vec { r } _ { 1 }$  of intersection of the plane  $\vec { r } \cdot \vec { n } = d$  and the perpendicular line  $\vec { r } = \vec { r } _ { 0 } + \hat { n }$  where  $t$  is a parameter given by -

  1. $\vec { r _ { 1 } } = \vec { r } _ { 0 } + \left( \dfrac { d - \vec { r } _ { 0 } \cdot \vec { n } } { \vec { n } ^ { 2 } } \right) \vec { n }$

  2. $\vec { r } _ { 1 } = \vec { r } _ { 0 } - \left( \dfrac { \vec { r } _ { 0 } \cdot \vec { n } } { \vec { n } ^ { 2 } } \right) \vec { n }$

  3. $\vec { r } _ { 1 } = \vec { r } _ { 0 } - \left( \dfrac { \vec { r } _ { 0 } \cdot \vec { n } - d } { | \vec { n } | } \right) \vec { n }$

  4. $\vec { r } _ { 1 } = \vec { r } _ { 0 } + \left( \dfrac { \vec { r } _ { 0 } \cdot \vec { n } } { | \vec { n } | } \right) \vec { n }$


Correct Option: A

If the line $\displaystyle \frac{x - 1}{1} = \frac{y + 1}{-2} = \frac{z + 1}{\lambda}$ lies in the plane $\displaystyle 3x - 2y + 5z = 0$ then $\displaystyle \lambda$ is

  1. $\displaystyle 1$

  2. $\displaystyle -\frac{7}{5}$

  3. $\displaystyle \frac{5}{7}$

  4. no possible value


Correct Option: B
Explanation:
We have equation of plane,
$3x-2y+5z=0.......(1)$

We have line,

$\dfrac{x-1}{1}=\dfrac{y+1}{-2}=\dfrac{z+1}{\lambda}=\mu......(2)$
General point on line is,
$P=(\mu+1,-2\mu-1,\lambda\mu-1)$
Since line (2) lies on line plane (1),so point P satisfy equation (1)
Therefore,
$3(\mu+1)-2(2\mu-1)+5(\lambda\mu-1)=0$
$3\mu+3+4\mu+2+5\mu\lambda-5=0$
$7\mu+5\mu\lambda=0$
$\lambda=\dfrac{-7}{5}$ 
Therefore option (B) is Correct.

The Foot of the $\displaystyle \perp$ from origin to the plane $\displaystyle 3x + 4y - 6z + 1 = 0$ is

  1. $\displaystyle - \frac {3}{61}, \frac {4}{61}, \frac {6}{61}$

  2. $\displaystyle \frac {-3}{61}, \frac {-4}{61}, \frac {-6}{61}$

  3. $\displaystyle \frac {4}{61}, \frac {-3}{61}, \frac {5}{61}$

  4. None of these


Correct Option: D
Explanation:

Clearly direction ratios of perpendicular drawn from origin to the given plane are $3,4,-6$
Hence equation of perpendicular line to the given plane and  passing through origin is given by,
$\cfrac{x}{3}=\cfrac{y}{4}=\cfrac{z}{-6}=k$ (say)
Now let foot of perpendicular be $P(3k, 4k, -6k)$
Also this point lie in the given plane $\Rightarrow 3(3k)+4(4k)-6(-6k)+1=0\Rightarrow k = -\cfrac{1}{61}$
Hence $P \equiv \left(-\cfrac{3}{61}, -\cfrac{4}{61}, \cfrac{6}{61}\right)$

The co-ordinate of a point where the line $(2, -3, 1)$ and $(3, -4, -5)$ cuts the plane $2x + y + z = 7$ are $(1, k, 7)$ then value of $k$ equals

  1. $1$

  2. $-2$

  3. $2$

  4. None of these


Correct Option: B
Explanation:

We know that the equation of the line passing through the point $(x _1,y _1,z _1) , (x _2,y _2,z _2) $


$ \dfrac{x-x _1}{x _2-x _1} = \dfrac {y-y _1}{y _2-y _1} =  \dfrac {z-z _1}{z _2-z _1}$

Now the given line passes through the points $(2,−3,1) , (3,−4,−5)$


Hence the equation of the line is $ \dfrac{x-2}{3-2} = \dfrac {y+3}{-4+3} =  \dfrac {z-1}{-5-1}$
`
$ \dfrac{x-2}{1} = \dfrac {y+3}{-1} =  \dfrac {z-1}{-6}$

Let the above equation be equal to :

$ \dfrac{x-2}{1} = 1$ .... (1)

$\dfrac {y+3}{-1} = a$ .....(2)

$\dfrac {z-1}{-6} = 7$ .....(3)

$\Rightarrow$ from eqn (2) 

$y=−4+a$

cube cuts the plane $2x+y+z=7$ ... (4)

substitute the $(1, k ,7)$ in eqn (4)

$2(1) + (-4+a) + 7 = 7$

$a = 2$

substitute a value in $y=−4+a$
we get $y = -4+2 = -2$

hence the ans will be $(1 , -2 , 7)$

The condition that the line $\displaystyle \frac{x-{\alpha }'}{l}=\frac{y -{\beta   }'}{m}=\frac{z-{\gamma  }'}{n}$ in the plane $Ax + By + Cz + D = 0$ is

  1. $A{\alpha }'+B{\beta }'+C{\gamma }'+D=0\ and\ Al+Bm+Cn\neq 0$

  2. $A{\alpha }'+B{\beta }'+C{\gamma }'+D\neq0\ and\ Al+Bm+Cn= 0$

  3. $A{\alpha }'+B{\beta }'+C{\gamma }'+D=0\ and\ Al+Bm+Cn= 0$

  4. $A{\alpha }'+B{\beta }'+C{\gamma }'=0\ and\ Al+Bm+Cn= 0$


Correct Option: C
Explanation:

The line $\dfrac{x- \alpha^1}{l}=\dfrac{y-\beta^1}{m}=\dfrac{z-\gamma^1}{n}$ in the plane $Ax+By+Cz+D=0$

then multiplication sum id corresponding direction ratio will be zero so here $Al+Bm+Cn=0$
 and  one thing more line passes through $(\alpha^1,\beta^1,\gamma^1)$
So, plane Also passe through $(\alpha^1,\beta^1,\gamma^1)$
hence $A\alpha^1+B\beta^1+C\gamma^1+D=0$

 Consider a point $P (1, 2, 3)$, plane $ \pi : x + y + z = 11 $ and the line $ L : \displaystyle \frac{x+1}{1}=\displaystyle \frac{y-12}{-2} = \displaystyle \frac{z-7}{2} $ The foot of the $ \perp  $ drawn from the point P meet the plane $ \pi $ at M, then co-ordinate of M is

  1. $ \left ( \displaystyle \frac{8}{3},:\displaystyle \frac{-11}{3},:\displaystyle \frac{14}{3} \right ) $

  2. $ \left ( \displaystyle \frac{-8}{3},:\displaystyle \frac{-11}{3},:\displaystyle \frac{-14}{3} \right ) $

  3. $ \left ( \displaystyle \frac{8}{3},:\displaystyle \frac{11}{3},:\displaystyle \frac{14}{3} \right ) $

  4. None of these


Correct Option: C
Explanation:
$P = (1,2,3) \rightarrow$ Point
$\Rightarrow\pi  : x+y+z = 11$
     $L : \dfrac { x+1 }{ 1 } =\dfrac { y-12 }{ -2 } =\dfrac { z-7 }{ 2 }$
Foot of perpendicular to the plane :
$\Rightarrow\dfrac { h-{ x } _{ 1 } }{ a } =\dfrac { k-{ y } _{ 1 } }{ b } =\dfrac { l-{ z } _{ 1 } }{ c } =\dfrac { -(a{ x } _{ 1 }+b{ y } _{ 1 }+c{ z } _{ 1 }) }{ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }$
     $\dfrac { h-1 }{ 1 } =\dfrac { k-2 }{ 1 } =\dfrac { l-3 }{ 1 } =\dfrac { 1+2+3 }{ 3 }$
     $h-1 = k-2= l-3= -2$
     $h= -1, k=0, l=1$
The required point is
$(h+\dfrac { 11 }{ 3 } , k+\dfrac { 11 }{ 3 } , l+\dfrac { 11 }{ 3 } )$
$(\dfrac { 8 }{ 3 } , \dfrac { 11 }{ 3 } ,\dfrac { 14 }{ 3 } )$
Hence the answer is $(\dfrac { 8 }{ 3 } , \dfrac { 11 }{ 3 } ,\dfrac { 14 }{ 3 } ).$

The point of intersection of the line $\dfrac{x-1}{3}=\dfrac{y+2}{4}=\dfrac{z-3}{-2}$ and the plane $2x-y+3z-1=0$, is

  1. $(-10, 10, 3)$

  2. $(10, 10, -3)$

  3. $(10, -10, 3)$

  4. $(10, -10, -3)$


Correct Option: B

Find the point where the line of intersection of the planes $x-2y+z=1$ and $x+2y-2z=5$ intersects the plane $3x+2y+z+6=0$.

  1. $P\left( 1,-2,-4 \right) $

  2. $P\left( 1,2,-4 \right) $

  3. $P\left( 1,-2,4 \right) $

  4. None of these


Correct Option: A
Explanation:

${ P } _{ 1 }\equiv x-2y+z=1$

${ P } _{ 2 }\equiv x+2y-2z=5$
Let $Dr's$ of intersection of planes be $a,b,c$
$\Rightarrow a-2b+c=0$ and $a+2b-2c=0$
$\displaystyle\Rightarrow \frac { a }{ 4-2 } =\frac { b }{ 1+2 } =\frac { c }{ 2+2 } \Rightarrow \frac { a }{ 2 } =\frac { b }{ 3 } =\frac { c }{ 4 } $
Calculating a point which lies on both plane ${P} _{1}$ and ${P} _{2}$,
$x-2y+z=1\Rightarrow2y=x+z-1$ and $x+2y-2z=5\Rightarrow2y=-x+2z+5$
$\Rightarrow x+z-1=-x+2z+5\Rightarrow 2x=z+6$
Now, $z=0\Rightarrow x=3\Rightarrow y=1$
$\therefore$ equation of line passing through $\left( 3,1,0 \right) $ and has $dr's$  $2,3,4$ is
$\displaystyle\therefore \frac { x-3 }{ 2 } =\frac { y-1 }{ 3 } =\frac { z }{ 4 } =\lambda $(say)
$\therefore$ general point on it is $P\left( 2\lambda +3,3\lambda +1,4\lambda  \right) $
Now, solving with plane $2x+2y+z+6=0$
$\Rightarrow 2\left( 2\lambda +3 \right) +2\left( 3\lambda +1 \right) +4\lambda +6=0\ \Rightarrow 4\lambda +6+6\lambda +2+4\lambda +6=0\ \Rightarrow 14\lambda +14=0\Rightarrow \lambda =-1$
$\Rightarrow P\left( 1,-2,-4 \right) $ is the required point of intersection.

The line joining the points $\left (2, -3, 1  \right )$ and $\left (3, -4, -5  \right )$ cuts a coordinate plane at the point.

  1. $\left (0, -1, 13 \right )$

  2. $\left ( 0, 0, 1 \right )$

  3. $\left ( -1, 0, 19 \right )$

  4. $\left ( 8, -9, 0 \right )$


Correct Option: A,C
Explanation:

Equation of the line is $\displaystyle \frac { x-2 }{ 2-3 } =\frac { y+3 }{ -3+4 } =\frac { z-1 }{ 1+5 } $ or $\displaystyle \frac { x-2 }{ -1 } =\frac { y+3 }{ 1 } =\frac { z-1 }{ 6 } $
Any point on the line $\left( -r+2,r-3,6r+1 \right) $ which cuts the $yz$-plane at the point where $-r+2=0$ or $r=2$
and the point of intersection is $\left( 0,-1,13 \right) $ 
Similarly it cuts the $zx$-plane at $\left( -1,0,19 \right) $ and $xy$ plane at $\displaystyle \left( \frac { 13 }{ 6 } ,\frac { -19 }{ 6 } ,0 \right) $.

Let line L: $\displaystyle \frac{x-1}{2} = \frac{y - 1}{1} = \frac{z - 0}{4} $ & Plane P: $x + 2y - z = 3$
Then which of the following is true?

  1. Line is perpendicular to plane

  2. Line is neither parallel nor perpendicular to plane

  3. Plane contains the line

  4. Line and plane do not intersect


Correct Option: C
Explanation:

Given line L: $\displaystyle \frac{x-1}{2} = \frac{y - 1}{1} = \frac{z - 0}{4} $

Plane P: $x + 2y - z = 3$
So from the below option c $i.e.$ plane contains the line is correct

If a line which passes through the point $A(0,\,1,\,2)$ and makes angle $\displaystyle\frac{\pi}{4},\,\displaystyle\frac{\pi}{4},\,\displaystyle\frac{\pi}{2}$ with $x,\,y,\,&amp;\,z$ axes respectively. The line meets the plane $x+y+z=0$ at point $B$. The length $\sqrt{2}AB$ is equal to

  1. $3$

  2. $-3$

  3. $4$

  4. $3 \sqrt {2}$


Correct Option: A
Explanation:

The D.C. of the line are $\begin{pmatrix}\displaystyle\frac{1}{\sqrt{2}},\,\displaystyle\frac{1}{\sqrt{2}},\,0\end{pmatrix}$
any point on the line at a distance $\lambda$ from $A(0,\,1,\,2)$
is $\begin{pmatrix}0+\displaystyle\frac{\lambda}{\sqrt{2}},\,1+\displaystyle\frac{\lambda}{\sqrt{2}},\,2+\lambda.0\end{pmatrix}$
which lies on $x+y+z=0$
$\therefore\;\displaystyle\frac{\lambda}{\sqrt{2}}+\begin{pmatrix}1+\displaystyle\frac{\lambda}{\sqrt{2}}\end{pmatrix}+2=0$
$\Rightarrow\;\displaystyle\frac{2\lambda}{\sqrt{2}}=-3\;\;\;\;\Rightarrow\;\;\;\;\lambda=\displaystyle\frac{-3}{\sqrt{2}}$
$\therefore\;B=\begin{pmatrix}-\displaystyle\frac{3}{2},\,-\displaystyle\frac{1}{2},\,2\end{pmatrix}$
$A=(0,\,1,\,2)\;&amp;\;B\equiv\;\begin{pmatrix}-\displaystyle\frac{3}{2},\,-\displaystyle\frac{1}{2},\,2\end{pmatrix}$
$AB=\sqrt{\displaystyle\frac{9}{4}+\displaystyle\frac{9}{4}}=\displaystyle\frac{3\sqrt{2}}{2}=\displaystyle\frac{3}{\sqrt{2}}$

The ratio in which the plane $\vec{r}.(\hat{i}-2\hat{j}+3\hat{k})=17$ divides the line joining the points $(-2\hat{i}+4\hat{j}+7\hat{k})$ and $(3\hat{i}-5\hat{j}+8\hat{k})$ is

  1. $1 : 5$

  2. $1 : 10$

  3. $3 : 5$

  4. $3 : 10$


Correct Option: D
Explanation:

Equation of plane in cartesian form is, $x-2y+3z-17=0.....(1)$
Assume this plane (1) divide the line segment joining the points $(-2,4,7)$ and $(3,-5,8)$ in $m:n$ ratio.
Therefore,  $\dfrac{m}{n} = \dfrac{-2-2(4)+3(7)-17}{3-2(-5)+3(8)-17}= \dfrac{-3}{10} < 0$
Hence plane (1) divides the given line segment externally  $3:10$.  

A straight line $\overrightarrow { r } =\overrightarrow { a } +\lambda \overrightarrow { b } $ meets the plane $\overrightarrow { r } .\overrightarrow { n } =0$ in $P$. The position vector of $P$ is

  1. $\displaystyle \overrightarrow { a } +\dfrac { \overrightarrow { a } .\overrightarrow { n }  }{ \overrightarrow { b } .\overrightarrow { n }  } \overrightarrow { b } $

  2. $\displaystyle \overrightarrow { a } -\dfrac { \overrightarrow { a } .\overrightarrow { n }  }{ \overrightarrow { b } .\overrightarrow { n }  } \overrightarrow { b } $

  3. $\displaystyle \dfrac { \overrightarrow { a } .\overrightarrow { n }  }{ \overrightarrow { b } .\overrightarrow { n }  } \overrightarrow { b } $

  4. None of these


Correct Option: B
Explanation:

A straight line $\overrightarrow { r } =\overrightarrow { a } +\lambda \overrightarrow { b } $ meets the plane $\overrightarrow { r } .\overrightarrow { n } =0$ at $P$ for which $\lambda$ is given by,

$\displaystyle \left( \overrightarrow { a } +\lambda \overrightarrow { b }  \right) .\overrightarrow { n } =0\Rightarrow \lambda =-\dfrac { \overrightarrow { a } .\overrightarrow { n }  }{ \overrightarrow { b } .\overrightarrow { n }  } $

Thus, the position vector of $P$ is

$\displaystyle \overrightarrow { r } =\overrightarrow { a } -\dfrac { \overrightarrow { a } .\overrightarrow { n }  }{ \overrightarrow { b } .\overrightarrow { n }  } \overrightarrow { b } $  $[$ putting the value of $\lambda$ in $\overrightarrow { r } =\overrightarrow { a } +\lambda \overrightarrow { b } ]$

 The value of $k$ such that $\displaystyle \dfrac{{x}-4}{1}=\dfrac{{y}-2}{1}=\dfrac{{z}-{k}}{2}$ lies in the plane $2x-4y+{z}=7$ is 

  1. $7$

  2. $-7$

  3. no real value

  4. $4$


Correct Option: A
Explanation:

Given equation of straight line $\displaystyle \dfrac { x-4 }{ 1 } =\dfrac { y-2 }{ 1 } =\dfrac { z-k }{ 2 } $


Since, the line lies in the plane $2x-4y+z=7$

$\therefore $ Point $\left( 4,2,k \right) $ must satisfy the plane.

$\Rightarrow 8-8+k=7 $

$\Rightarrow k=7$

The plane $x-2y+z-6=0$ and the line $\displaystyle\frac{x}{1}=\displaystyle\frac{y}{2}=\displaystyle\frac{z}{3}$ are related as.

  1. Parallel to the plane

  2. At right angle to the plane

  3. Lies in the plane

  4. Meets the plane obliquely


Correct Option: A
Explanation:

Given line is $\displaystyle\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ any point on the line is $(t, 2t, 3t)$. It lies in the given plane, if $t-2(2t)+3t-6=0$
i.e., $0\cdot t=6$, which is not true for any real $t$. So, the line and plane do not meet. i.e., the line is parallel to the plane.

The plane ax + by + cz = 1 meets the coordinate axes in A, B and C. The centroid of $\triangle ABC$ is

  1. $(3a, 3b, 3c)$

  2. $(\dfrac{a}{3}, \dfrac{b}{3}, \dfrac{c}{3})$

  3. $(\dfrac{3}{a}, \dfrac{3}{b}, \dfrac{3}{c})$

  4. $(\dfrac{1}{3a}, \dfrac{1}{3b}, \dfrac{1}{3c})$


Correct Option: D
Explanation:
 Coordinates of A,B,C are $(\frac{1}{a},0),(\frac{1}{b},0),(\frac{1}{c},0) $respectively
  centriod of $ \Delta ABC = (\frac{1}{3a},\frac{1}{3b},\frac{1}{3c})$

The plane $\frac{x}{y}+\frac{y}{3}+\frac{z}{4}$ =1 cutes the axes in A,B,C, then the are of the $\Delta ABC$ is;

  1. $\sqrt{29}$

  2. $\sqrt{41}$

  3. $\sqrt{61}$

  4. None of these


Correct Option: C
Explanation:

We have,

$\dfrac{x}{2}+\dfrac{y}{3}+\dfrac{z}{4}=1$

Compare this equation of plane,

$\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$

Then,

$a=2,\,b=3,\,c=4$

Then,

The coordinate of X-axis is $=A\left( a,0,0 \right)$$=A\left( 2,0,0 \right)$

The coordinate of Y-axis is $=B\left( 0,b,0 \right)$$=A\left( 0,3,0 \right)$

The coordinate of X-axis is $=C\left( 0,0,c \right)$$=C\left( 0,0,2 \right)$

We know that, the area of  \[\Delta ABC\] is

$ Area\,of\,\Delta ABC=\dfrac{1}{2}\sqrt{{{a}^{2}}{{b}^{2}}+{{b}^{2}}{{c}^{2}}+{{c}^{2}}{{a}^{2}}} $

$ =\dfrac{1}{2}\sqrt{{{2}^{2}}\times {{3}^{2}}+{{3}^{2}}\times {{4}^{2}}+{{4}^{2}}\times {{2}^{2}}} $

$ =\dfrac{1}{2}\sqrt{36+144+64} $

$ =\dfrac{1}{2}\sqrt{180+64} $

$ =\dfrac{1}{2}\sqrt{244} $

$ =\dfrac{1}{2}\sqrt{2\times 2\times 61} $

$ =\sqrt{61} $

Hence, this is the answer.

Perpendicular is drawn from the point $(0,3,4)$ to the plane $2x -2y + z + (-10) = 0$, then co-ordinates of the foot of the L's are

  1. $\displaystyle \left ( \frac{8}{3},\frac{1}{3},\frac{16}{3}\right )$

  2. $\displaystyle \left ( -\frac{8}{3},\frac{1}{3},\frac{16}{3}\right )$

  3. $\displaystyle \left ( \frac{8}{3},-\frac{1}{3},\frac{16}{3}\right )$

  4. $\displaystyle \left ( \frac{8}{3},\frac{1}{3},-\frac{16}{3}\right )$


Correct Option: A
Explanation:

We have point $(0,3,4)$ and we have direction's ratio also of line so we can write equation of line
$\dfrac{x-0}{2}=\dfrac{y-3}{-2}=\dfrac{z-4}{1}=k$
So consider a point which is lied on line $(2k,-2k+3,k+4)$
This point lies on plane also so it will satisfy the equation of plane:
$\Rightarrow 2(2k)-2(-2k+3)+k+4-10=0$
$\Rightarrow k=\dfrac{4}{3}$
Substitute the value of $k$ in point $\left (\dfrac{8}{3},\dfrac{1}{3},\dfrac{16}{3}\right)$

Let the line $\displaystyle \frac{x-2}{3}= \frac{y-1}{-5}= \frac{z+2}{2}$ lie in the plane $x+3y-\alpha z+\beta = 0$. Then $\left ( \alpha ,\beta  \right )$ equals :

  1. $\left ( -6,7 \right )$

  2. $\left ( 5,-15 \right )$

  3. $\left ( -5,5 \right )$

  4. $\left ( 6,-17 \right )$


Correct Option: A
Explanation:

The line is $\displaystyle \frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}2{}$


The direction ratios of the line are $(3,-5,2)$

As the line is in the plane $x+3y-az+ \beta =0$,

We have $\left ( 3 \right )\left ( 1 \right )+\left ( -5 \right )\left ( 3 \right )+2\left ( -\alpha  \right )=0$

$\Rightarrow-12-2 \alpha =0$

$ \therefore \alpha = -6$

Again $(2,1,-2)$ lies on the plane

$\Rightarrow 2+3+2 \alpha + \beta =0$

$\Rightarrow \beta = -2 \alpha -5=12-5=7$

Hence, $\left ( \alpha ,\beta  \right )$ is $\left ( -6,7 \right )$

The line $x -2y + 4z + 4 = 0$, $x + y + z - 8 = 0$ intersects the plane $x - y + 2z + 1 = 0$ at the point

  1. $\left ( 3, 2, 3 \right )$

  2. $\left ( 5, 2, 1 \right )$

  3. $\left ( 2, 5, 1 \right )$

  4. $\left ( 3, 4, 1 \right )$


Correct Option: C
Explanation:

Given  lines
 $x -2y + 4z + 4 = 0$    ....(1)
 $x + y + z - 8 = 0$       .....(2)
Subtracting (2) from (1), we get
$\Rightarrow y-z=4$        .....(3)
Given equation of plane $x-y+2z+1=0$
Since, the line intersects the plane, so using (3), we get 
$\Rightarrow x+z=3$      ......(4)
Hence, $y=5$, $z=1 $ and $x=2$
Hence, option C is correct.

$L: \displaystyle \frac{x\, +\, 1}{2}= \frac{y\, +\, 1}{3}= \frac{z\, +\, 1}{4}$
$\pi _{1}:\, x\, +\, 2y\, +\, 3z= 14,\, \pi _{2}:\, 2x\, -\, y\, +\, 3z= 27$

If the line $L$ meets the plane $\pi _{1}$ in the point $P$, and the coordinates of $P$ are $\left ( \alpha ,\, \beta ,\, \gamma  \right )$, then $\alpha ^{2}\, +\, \beta ^{2}\, +\, \gamma ^{2}$ is equal to

  1. $3$

  2. $14$

  3. $28$

  4. $29$


Correct Option: B
Explanation:
$L=\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z+1}{4}$     ........(i) and plane
$\pi \Rightarrow x+2y+3z=14$        .......(ii)
Given that $L$ meets plane $\pi _1$ (means intersection points) so from $eq^n$ (i)
$\Rightarrow \dfrac{x+1}{2}=\dfrac{y+1}{3}=\dfrac{x+1}{4}=K$
$\Rightarrow x= 2k-1\,\, and\,\, y=3k-1,z=4k-1  $
putting this values in ..... (ii)
So $(2k-1)+2(3k-1)+3(4k-1)=14$
$\Rightarrow 2k-1+6k-2+12k-3=14$
$20k-6=14$
$\Rightarrow 20k=20 \rightarrow k=1$
so points $x=1,y=2,z=3$ inform of $\alpha ,\beta,\gamma= \alpha =1,\beta=2,\gamma =3$
so $\alpha^2+\beta^2+\gamma^2=1^2+2^2+3^2$
$=14$

A line with positive direction cosines passes through the point $\displaystyle P\left ( 2,-1,2 \right )$ and makes equal angles with the coordinates axis. The line meet the plane $\displaystyle 2x+y+z=9$ at ponit $Q$.
The length of the line segment $PQ$ equals.

  1. $1$

  2. $\displaystyle \sqrt{2}$

  3. $\displaystyle \sqrt{3}$

  4. $2$


Correct Option: C
Explanation:

Equation of a line through the point $\displaystyle P\left ( 2,-1,2 \right )$, equally inclined to the axes is given by,
$\displaystyle \frac{x-2}{1}=\frac{y+1}{1}=\frac{z-2}{1}=k$ (say)
Any point on the line is $Q ( k+2,k-1, k+2  )$ which lies on the plane $\displaystyle 2x+y+z=9$
 $\Rightarrow \displaystyle 2\left ( k+2 \right )+k-1+k+2=9 \Rightarrow k=1$,
For this values of $k$, the coordinates of $Q$ are $\displaystyle \left ( 3,0,3 \right )$.
So, $\displaystyle PQ=\sqrt{\left ( 3-2 \right)^2+\left ( 0+1 \right )^2+\left ( 3-2 \right )^2}=\sqrt{3}$

Find the point where the line of intersection of the planes $ x - 2y + z = 1$ and $x + 2y - 2z = 5$, intersects the plane $2x + 2y + z + 6 = 0$

  1. $(1, -2, -4)$

  2. $(0,0,-6)$

  3. $(1,0,-8)$

  4. $(-1,-1,-2)$


Correct Option: A

The line passing through the points $(5, 1,  a)$ and $(3, b, 1)$ crosses the $yz$-plane at the point $\left (0,\dfrac{17}{2},\dfrac{-13}{2}\right)$. Then,

  1. $a = 2, b = 8$

  2. $a = 4, b = 6$

  3. $a = 6, b = 4$

  4. $a = 8, b = 2$


Correct Option: C
Explanation:

Equation of line passing through $(5, 1, a)$ and $(3, b, 1)$ is
$\dfrac{x-3}{5-3}= \dfrac{y-b}{1-b}= \dfrac{z-1}{a-1}$   ...(i)


Point $\left ( 0, \dfrac{17}{2}, -\dfrac{13}{2} \right )$ satisfies equation (i), we get

$-\dfrac{3}{2} = \dfrac{\dfrac{17}{2} -b}{1-b} = \dfrac{-\dfrac{13}{2}-1}{a-1}$

$\Rightarrow  a-1 = \dfrac{\left ( -\dfrac{15}{2} \right )}{\left ( -\dfrac{3}{2} \right )} = 5$
$\Rightarrow  a = 6$


Also,  $-3\left ( 1 - b \right )= 2 \left ( \dfrac{17}{2} - b\right )$

$\Rightarrow  3b - 3 = 17 - 2b$

$\Rightarrow  5b = 20   $

$  \Rightarrow  b = 4$

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