Tag: pair of straight lines

Questions Related to pair of straight lines

The pair of lines represented by $\displaystyle :3ax^{2}+5xy+\left ( a^{2}-2 \right )y^{2}= 0$ and at right angles to each other, then value $ \left ( s \right )$ of $a$ is/are:

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. $\displaystyle :\frac{-3+\sqrt{17}}{2}$

  2. $\displaystyle :\frac{-3-\sqrt{17}}{2}$

  3. $\displaystyle :\frac{3+\sqrt{17}}{2}$

  4. $\displaystyle :\frac{3-\sqrt{17}}{2}$

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Correct Option: A,B
Explanation:

The pair of lines represented by $\displaystyle :3ax^{2}+5xy+\left ( a^{2}-2 \right )y^{2}= 0$ and at right angles to each other.
pair of lines given by $ax^2+2hxy+by^2=0$ are at right angles if $a+b=0$
$\Rightarrow 3a+a^2-2=0$
$\Rightarrow a^2+3a-2=0$
$\therefore a=\displaystyle\frac{-3\pm \sqrt{17}}{2}$
Hence, options A and B.

If one of the line given by the equation $a _{1}x^{2}+2h _{1}xy+b _{1}y^{2}=0$ coincides with one of the lines given by $a _{2}x^{2}+2h _{2}xy+b _{2}y^{2}=0$ and the other lines represented by them be perpendicular then $\dfrac {h _{1}a _{2}b _{2}}{b^{2}-a _{2}}\dfrac {h _{2}a _{1}b _{1}}{b _{1}-a _{1}}=\dfrac {1}{2}\sqrt {-a _{1}a _{2}b _{1}b _{2}}$.

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. True

  2. False

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Correct Option: A

The triangle formed by the lines whose combined equation is  $\displaystyle (y^{2}-4xy-x^{2}) ( x+y-1  )=0$ is

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. equilateral.

  2. right angled.

  3. isosceles.

  4. obtuse angled.

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Correct Option: B
Explanation:

Given equation of the pair of straight lines is  $x^{2}-4xy-y^{2}=0$

Since, coefficient of $x^2$ + coefficient of $y^2$ $= 0$ 

The two lines are perpendicular.

Therefore, triangle formed is right angled.

Find the equation of the line perpendicular to $x-7y+5=0$ and having x-intercept 3.

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. $x+y-21=0$

  2. $x-7y+21=0$

  3. $7x+y-21=0$

  4. None of these

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Correct Option: C
Explanation:

The equation of a line perpendicular to $x-7y+5=0$ is $7x+y+\lambda =0$.


Its x-intercept is 3. This means that the line cuts x-axis at a distance of 3 units from the origin. Consequently, it passes through the point (3,0) on x-axis.


Therefore, $21+0+\lambda =0$

$\lambda = -21$

Thus, the equation of the required line is $7x+y-21=0$.

If $2x^{2}+3xy+my^{2}=0$ represents two real and mutually perpendicular lines then $m$ is

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. any negative real number

  2. any positive real number

  3. $-2$

  4. none of these

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Correct Option: C
Explanation:

From above formula
$\tan90^{\circ}=\left | \dfrac{2\sqrt{(\frac{3}{2})^{2}+2m}}{2+m} \right |=\dfrac{1}{0}$
$m=-2$

The product of the perpendiculars from origin to the pair of lines $ a x ^ { 2 } + 2 h x y + b y ^ { 2 } + 2 g x + 2 f y + c = 0 $ is

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. $

    \frac { | c | } { \sqrt { ( a + b ) ^ { 2 } + 4 h ^ { 2 } } }

    $

  2. $

    \frac { | c | } { \sqrt { ( a + b ) ^ { 2 } - 4 h ^ { 2 } } }

    $

  3. $

    \frac { | c | } { \sqrt { ( a - b ) ^ { 2 } + 4 h ^ { 2 } } }

    $

  4. $

    \frac { | c | } { \sqrt { ( a - b ) ^ { 2 } - 4 h ^ { 2 } } }

    $

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Correct Option: A

The equation $\displaystyle ax^{3}-9yx^{2}-y^{2}x+4y^{3}=0 $ represents three straight lines. If two of the lines are perpendicular to each other, then the value of $a$ is:

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. 5

  2. -5

  3. 4

  4. -4

Show answer
Correct Option: A,D
Explanation:

The given equation is $ax^3 - 9yx^2 -y^2x + 4y^3 = 0$, which represents three straight lines.


We can see that all the lines passes from origin $(0,0)$

Let's assume the lines are given by equation $y = mx$

Putting $y = mx$ in the given equation of three straight lines, we get,

$\Rightarrow ax^3- 9(mx)(x^2) - (mx)^2x + 4(mx)^3 = 0$

$\Rightarrow (a - 9m - m^2 + 4m^3) x^3 = 0$

$\Rightarrow 4m^3 -m^2 -9m +a = 0$ ....$(1)$

This equation in $m$ has three roots, $m _1$, $m _2$ and $m _3$, which are three slopes of three lines respectively.

If the two lines are perpendicular then let's assume $m _1.m _2 = -1$,

Product of roots in equation $(1)$ is $m _1.m _2.m _3 = \dfrac{-a}{4}$

Hence $m _3 = \dfrac{a}{4}$

also from equation $(1)$, $m _1 + m _2 + m _3 = \dfrac{1}{4}$

$\Rightarrow m _1m _2 + m _3(m _1 + m _2) = \dfrac{-9}{4}$

$\Rightarrow m _1 + m _2 = \dfrac{1-a}{4}$

$\Rightarrow a(1-a) = -20$

By Solving the above equation we get $a = 5, -4$

Equation $\displaystyle ax^{3}-9yx^{2}-y^{2}x+4y^{3}=0$ represents three straight lines. If two of the lines are perpendicular to each other then the value of a is

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. 5

  2. -5

  3. 4

  4. -4

Show answer
Correct Option: A,D
Explanation:

The given equation is $ax^3 - 9yx^2 -y^2x + 4y^3 = 0$, which represents three straight lines.


We can see that all the lines passes from origin $(0,0)$

Let's assume the lines are given by equation $y = mx$

Putting $y = mx$ in the given equation of three straight lines, we get,

$\Rightarrow ax^3- 9(mx)(x^2) - (mx)^2x + 4(mx)^3 = 0$

$\Rightarrow (a - 9m - m^2 + 4m^3) x^3 = 0$

$\Rightarrow 4m^3 -m^2 -9m +a = 0$ ....$(1)$

This equation in $m$ has three roots, $m _1$, $m _2$ and $m _3$, which are three slopes of three lines respectively.

If the two lines are perpendicular then let's assume $m _1.m _2 = -1$,

Product of roots in equation $(1)$ is $m _1.m _2.m _3 = \dfrac{-a}{4}$

Hence $m _3 = \dfrac{a}{4}$

also from equation $(1)$, $m _1 + m _2 + m _3 = \dfrac{1}{4}$

$\Rightarrow m _1m _2 + m _3(m _1 + m _2) = \dfrac{-9}{4}$

$\Rightarrow m _1 + m _2 = \dfrac{1-a}{4}$

$\Rightarrow a(1-a) = -20$

By Solving the above equation we get $a = 5, -4$

The pair of lines represented by $3ax^{2}+5xy+\left ( a^{2}-2 \right )y^{2}= 0$ and $\perp $ to each other for

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. two values of $a$

  2. for all $a$

  3. for one value of $a$

  4. for no values of $a$

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Correct Option: A
Explanation:

 Using fact: Pair of lines $\displaystyle Ax^{2}+2hxy+By^{2}=0$ are 


$\displaystyle \perp $ to each other if $\displaystyle A+B=0$ 

$\displaystyle \Rightarrow 3a+a^{2}-2=0 $ $\displaystyle \Rightarrow a^{2}+3a-2=0 $ $\displaystyle\Rightarrow $ There exist two value of a as $\displaystyle D> 0$

The equation $ \displaystyle 3x^{2}-8xy-3y^{2}=0 $ and $ \displaystyle x-2y=3 $ represents the sides of a triangle which is

business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

  1. equilateral

  2. isosceles

  3. right angled triangle

  4. none of these

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Correct Option: C
Explanation:

Given equation of pair of lines $\displaystyle 3{x}^{2}-8xy-3{y}^{2}=0$,
Let $\displaystyle \frac{y}{x}=m$, we get $\displaystyle 3{m}^{2}+8m-3=0$ Let $\displaystyle {m} _{1}$ and ${m} _{2} $ are slopes of lines,we get $\displaystyle {m} _{1}\times{m} _{2}=-1$

$\displaystyle \therefore $ lines are perpendicular to each other.
Then triangle is Right angled triangle .