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Straight lines-reduction into various forms - class-XI

Description: straight lines-reduction into various forms
Number of Questions: 20
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Tags: maths coordinates, points and lines coordinate geometry locus and straight line straight lines mathematics and statistics the straight line straight line two dimensional analytical geometry
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If the line $(2x+y+1)+\lambda(x-y+1)=0$ is parallel to $y-axis$ then value of $\lambda$ is ?
mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $1$

  2. $-1$

  3. $\dfrac{1}{2}$

  4. $2$

Show answer
Correct Option: A
Explanation:
Parallel to y-axis $\Rightarrow y=0$
$2x+y+1+\lambda x-\lambda y+\lambda =0$
$x(2+\lambda )+y(1-\lambda )+(1+\lambda )=0$
coefficient of $y=0$
$1-\lambda =0\Rightarrow \lambda =1$

The equation of the line passing through $(-4, 3)$, parallel to the $3x+7y+6=0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. 3x+7y-9=0

  2. 3x+7y+9=0

  3. 3x+7y+3=0

  4. 3x+7y+12=0

Show answer
Correct Option: A
Explanation:

The line parallel to $3x+7y+6=0$ is $3x+7y+k=0$

It passes through $(-4,3)$
$\implies 3(-4)+7(3)+k=0\-12+21+k=0\\implies k=-9$
So the required equation is $3x+7y-9=0$

The slope and  the y-intercept  of the given line, $y-3x -6=0$ are respectively,

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $3, -6$

  2. $-3, -6$

  3. $3, 6$

  4. $-3, 6$

Show answer
Correct Option: C
Explanation:

The given equation is $y-3x-6=0$ ........ $(1)$


To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 


$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,

$y-3x-6=0\implies y=3x+6$

Comparing it with $y=mx+c$ we get,

slope$=m=3$ and y-intercept$=c=6$

Hence, option C is correct.

The slope and y-intercept of the following line are respectively

$2y + 2x - 5 = 0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $ slope=m=1\quad and\quad y-intercept=c=\frac { 5 }{ 2 } . $

  2. $ slope=m=1/5\quad and\quad y-intercept=c=\frac { 2 }{ 5 } . $

  3. $ slope=m=-1\quad and\quad y-intercept=c=\frac { 5 }{ 2 } . $

  4. $ slope=m=-1/5\quad and\quad y-intercept=c=\frac { 2 }{ 5 } . $

Show answer
Correct Option: C
Explanation:

The given equation is $2y+2x-5=0$ ........ $(1)$

To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 
$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,
$2y+2x-5=0\implies y=-x+\dfrac{5}{2}$
Comparing it with $y=mx+c$ we get,
slope$=m=-1$ and y-intercept$=c=\dfrac{5}{2}$
Hence, option C is correct.

The slope and y-intercept of the following line are respectively

$7x-y + 3 =0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $ slope=m=7/3\quad and\quad y-intercept=1.\ $

  2. $ slope=m=-7\quad and\quad y-intercept=3.\ $

  3. $ slope=m=-7/3\quad and\quad y-intercept=1.\ $

  4. $ slope=m=7\quad and\quad y-intercept=3.\ $

Show answer
Correct Option: D
Explanation:

The given equation is $7x-y+3=0$ ........ $(1)$

To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 
$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,
$7x-y+3=0\implies y=7x+3$
Comparing it with $y=mx+c$ we get,
slope$=m=7$ and y-intercept$=c=3$
Hence, option D is correct.

The slope and  the y-intercept  of the given line, $2x-3y = 7$ are respectively,

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $\dfrac{3}{2}, \dfrac{-3}{7}$

  2. $\dfrac{2}{3}, \dfrac{-7}{3}$

  3. $\dfrac{3}{2},  \dfrac{3}{7}$

  4. $\dfrac{2}{3}, \dfrac{7}{3}$

Show answer
Correct Option: B
Explanation:

The given equation is $2x-3y=7$ ........ $(1)$


To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is, 

$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,

$2x-3y=7\implies 3y=2x-7\implies y=\dfrac{2}{3} x+\left( -\dfrac{7}{3}  \right)$

Comparing it with $y=mx+c$ we get,

slope$=m=\dfrac{2}{3}$ and y-intercept$=c=-\dfrac{7}{3}$

Hence, option B is correct.

The slope and y-intercept of the following line are respectively

$4x-y=0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $ slope=m=4\quad and\quad y-intercept=0.\ $

  2. $ slope=m=-4\quad and\quad y-intercept=0.\ $

  3. $ slope=m=1/4\quad and\quad y-intercept=0.\ $

  4. $ slope=m=0\quad and\quad y-intercept=1/4.\ $

Show answer
Correct Option: A
Explanation:

The given equation is $4x-y=0$ ........ $(1)$

To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 
$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,
$4x-y=0\implies y=4x$
Comparing it with $y=mx+c$ we get,
slope$=m=4$ and y-intercept$=c=0$
Hence, option A is correct.

The slope and y-intercept of the following line are respectively

$8x-4y-1=0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $ slope=m=\frac { -1 }{ 2 } \quad and\quad y-intercept=\frac { 1 }{ 4 } . $

  2. $ slope=m=2 \quad and\quad y-intercept=-\frac { 1 }{ 4 } . $

  3. $ slope=m=-\frac { 1 }{ 2 } \quad and\quad y-intercept=-\frac { 1 }{ 4 } . $

  4. $ slope=m=\frac { 1 }{ 2 } \quad and\quad y-intercept=\frac { 1 }{ 4 } . $

Show answer
Correct Option: B
Explanation:
Given line
$8x-4y-1=0$
Comparing above eq with $y=mx+c$ where m is slope and c is y intercept
Here $m=2,c=-\dfrac{1}{4}$

The slope and y-intercept of the following line are respectively

$5x - 2y = 3$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $ slope=m=-\frac { 5 }{ 2 } \quad and\quad y-intercept=-\frac { 3 }{ 2 } . $

  2. $ slope=m=\frac { 5 }{ 2 } \quad and\quad y-intercept=\frac { 3 }{ 2 } . $

  3. $ slope=m=\frac { 5 }{ 2 } \quad and\quad y-intercept=-\frac { 3 }{ 2 } . $

  4. $ slope=m=-\frac { 5 }{ 2 } \quad and\quad y-intercept=\frac { 3 }{ 2 } . $

Show answer
Correct Option: C
Explanation:

The given equation is $5x-2y=3$ ........ $(1)$

To obtain the slope and $y-$intercept of the given equation, we write it in slope-intercept form which is 
$y=mx+c$, where $m$ and $c$ are slope and $y-$intercept

From $(1)$,
$5x-2y=3\implies y=\dfrac{5}{2}x - \dfrac{3}{2}$
Comparing it with $y=mx+c$ we get,
slope$=m=\dfrac{5}{2}$ and y-intercept$=c=-\dfrac{3}{2}$
Hence, option C is correct.

The slope and $y$-intercept of the following line are respectively

$5x-8y =-2$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope $=m=-\dfrac { 5 }{ 8 } $ and $ y$-intercept $=\dfrac { 1 }{ 4 }  $

  2. slope $=m=\dfrac { 5 }{ 8 } $ and $y$-intercept $=-\dfrac { 1 }{ 4 }  $

  3. slope $=m=-\dfrac { 5 }{ 8 }$ and $ y$-intercept $=-\dfrac { 1 }{ 4 }  $

  4. slope $=m=\dfrac { 5 }{ 8 } $ and $ y$-intercept $=\dfrac { 1 }{ 4 }  $

Show answer
Correct Option: D
Explanation:

$ To\quad obtain\quad the\quad slope\quad and\quad y-intercept\quad of\quad an\quad equation\quad of\quad any\quad form\ write\quad it\quad in\quad slope-intercept\quad form\quad which\quad is\quad y=mx+c.\ Then\quad slope=m\quad and\quad y-intercept=c.\  $
The given equation is: $5x - 8y = -2$
$-8y = -5x - 2$
$y = \dfrac{5}{8}x + \dfrac{2}{8} $
Compare it with general form of equation: $y = mx + c$
m = $\dfrac{5}{8}$, y - intercept = c = $  \dfrac{1}{4}$

For the equation given below, find the the slope and the y-intercept : $\displaystyle 3y=7$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $\displaystyle 0 \ and \ \frac{7}{3}$

  2. $\displaystyle 0 \ and \ -\frac{7}{3}$

  3. $\displaystyle -\frac{7}{3} \ and \ 0$

  4. $\displaystyle \frac{7}{3} \ and \ 0$

Show answer
Correct Option: A
Explanation:

The equation of any straight line can be written as $ y =

mx + c $, where $m$ is its slope and $c$ is its y - intercept.
$ 3y = 7 $ can be written as $ y = \frac {7}{3} $

Comparing this equation with the standard form of the equation, we get:
$ m = 0 , c =  \frac {7}{3} $

Hence, slope of $ 3y = 7 $ is $ 0 $  and y -intercept is $   \frac {7}{3} $

$ax + by + c = 0$ does not represent an equation of line if ____.

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $a = c = 0, b \neq 0$

  2. $b = c = 0, a \neq 0$

  3. $a = b = 0$

  4. $c = 0, a \neq 0, b \neq 0 $

Show answer
Correct Option: C
Explanation:

$ax+by+c=0$ will represent the equation of line If both or one coefficient of $x$ and $y$ is not equal to $0$.

Therefore, if $a=b=0$ then it will not represent the equation of a line.

Find slope, x-intercept & y-intercept of the line 2x - 3y + 5 = 0

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $\dfrac{-5}{2},\dfrac{5}{3},\dfrac{2}{3}$

  2. $\dfrac{-5}{2},\dfrac{5}{3},\dfrac{1}{3}$

  3. $\dfrac{-3}{2},\dfrac{5}{3},\dfrac{2}{3}$

  4. $\dfrac{-5}{2},\dfrac{4}{3},\dfrac{2}{3}$

Show answer
Correct Option: A

Find the slope and $y$-intercept of the line $2x + 5y = 1$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope $=$ $-\dfrac{2}{5}$, $y$-intercept $=$ $\dfrac{1}{5}$

  2. slope $=$ $-\dfrac{1}{5}$, $y$-intercept $=$ $\dfrac{1}{5}$

  3. slope $=$ $-\dfrac{2}{3}$, $y$-intercept $=$ $\dfrac{1}{5}$

  4. slope $=$ $-\dfrac{2}{5}$, $y$-intercept $= $ $\dfrac{2}{5}$

Show answer
Correct Option: A
Explanation:
The slope intercept form of the line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $2x+5y=1$ in slope intercept form:
$2x+5y=1$
$5y=-2x+1$
$y=-\dfrac { 2 }{ 5 } x+\dfrac { 1 }{ 5 }$ 
Hence, the slope of the line $2x+2y=-2$ is $m=-\dfrac { 2 }{ 5 }$ and the $y$-intercept is $\dfrac { 1 }{ 5 }$.

Find the slope and $y$-intercept of the line $-5x + y = 5$.

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope $= 5, y$-intercept $= -5$

  2. slope $= 5, y$-intercept $= -4$

  3. slope $= 5, y$-intercept $= 5$

  4. slope $= 5, y$-intercept $= -1$

Show answer
Correct Option: C
Explanation:

The slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. 

Given straight line $-5x+y=5$ can be written as, $y=5x+5$
Now comparing above equation with slope-intercept form $y=mx+c$

We get, slope $=m = 5$ and $y$-intercept $=c=5$.

Hence, option C is correct.

Find the slope and $y$-intercept of the line $0.2x - y = 1.2$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope $= 0.2$, $y$-intercept $= -1.2$

  2. slope $= 1.2$, $y$-intercept $= -1.2$

  3. slope $= 0.2$, $y$-intercept $= -2.2$

  4. slope $= 0.2$, $y$-intercept $= -1.3$

Show answer
Correct Option: A
Explanation:
The slope intercept form of the line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $0.2x-y=1.2$ in slope intercept form:
$0.2x-y=1.2$
$\Rightarrow -y=-0.2x+1.2$
$\Rightarrow y=0.2x-1.2$
Hence, the slope of the line $0.2x-y=1.2$ is $m=0.2$ and the $y$-intercept is $-1.2$.

Find the slope and $y$-intercept of the line $2x + 2y = -2$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope = 1, y-intercept $= -3$

  2. slope = -1, y-intercept $= -1$

  3. slope = 1, y-intercept $= 3$

  4. slope = 1, y-intercept $= 1$

Show answer
Correct Option: B
Explanation:
The slope intercept form of the line is $y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $2x+2y=-2$ in slope intercept form:
$2x+2y=-2$
$\Rightarrow 2y=-2x-2$
$\Rightarrow y=-x-1$
Hence, the slope of the line $2x+2y=-2$ is $m=-1$ and the $y$-intercept is $-1$.

Find the slope and $y$-intercept of the line $x - y = 3$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. slope $= 2$, $y$-intercept $= -3$

  2. slope $= 0$, $y$-intercept $= -3$

  3. slope $= 1$, $y$-intercept $= -3$

  4. slope $= 1$, $y$-intercept $= 3$

Show answer
Correct Option: C
Explanation:
The slope intercept form of the line is $y=mx+c$ where $m$ is the slope of the line and $c$ is the $y$-intercept.
Change the equation $x-y=3$ in slope intercept form:
$x-y=3$
$\Rightarrow -y=-x+3$
$\Rightarrow y=x-3$
Hence, the slope of the line $x-y=3$ is $m=1$ and the $y$-intercept is $-3$.

A line in the $xy$-plane passes through the origin and has a slope of $\dfrac{1}{7}$. Which of the following points lies on the line?

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $\left(0, 7\right)$

  2. $\left(1, 7\right)$

  3. $\left(7, 7\right)$

  4. $\left(14, 2\right)$

Show answer
Correct Option: D
Explanation:

If any straight line passes through origin, then it must of the form $y = mx$.


Now if the slope is $\dfrac{1}{7}$, then line will be $y = \dfrac{1}{7}x \ $ or $ \ 7y -x = 0$

We can see out of all the points only point $(14,2)$ satisfies the equation of the line. Hence Only $(14,2)$ lies on the line.

Correct option is $D$

Find an equation of the line through the points $(-3,5)$ and $(9,10)$ and write it in standard form $Ax+By=C$, with $A>0$

mathematics and statistics coordinates, points and lines general equation of a line: ax+by+c=0 general equation of a line reducing equation of straight line to standard form

  1. $6x-10y=-75$

  2. $5x-12y=-75$

  3. $4x-11y=-65$

  4. $x-6y=-15$

Show answer
Correct Option: B
Explanation:
Given points are $(-3,5)$ and $(9,10)$
The slope of the line is given by:
$m =\dfrac{ (10-5)}{[9-(-3)] }= \dfrac {5}{12}$
The equation becomes: 
$y - 5 = \left (\dfrac {5}{12}\right) [x-(-3)]$ 
$y-5=\left (\dfrac {5}{12}\right)(x+3)$ 
Solve it and get the equation- 
$12y-60=5x+15$ 
$5x-12y=-75$ 

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