Tag: condition for perpendicular and coincident lines and bisectors of angles

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

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:

  1. $\displaystyle \:\frac{-3+\sqrt{17}}{2}$
  2. $\displaystyle \:\frac{-3-\sqrt{17}}{2}$
  3. $\displaystyle \:\frac{3+\sqrt{17}}{2}$
  4. $\displaystyle \:\frac{3-\sqrt{17}}{2}$
Reveal answer Fill a bubble to check yourself
A,B Correct answer
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.

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

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 $\dfrac {h _{1}a _{2}b _{2}}{b^{2}-a _{2}}\dfrac {h _{2}a _{1}b _{1}}{b _{1}-a _{1}}=\dfrac {1}{2}\sqrt {-a _{1}a _{2}b _{1}b _{2}}$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a known identity in coordinate geometry regarding the intersection and perpendicularity of lines represented by homogeneous second-degree equations. The statement provided is a standard theorem result.

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

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

  1. equilateral.

  2. right angled.

  3. isosceles.

  4. obtuse angled.
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

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

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

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

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $7x+y-21=0$.

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

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

  1. any negative real number

  2. any positive real number
  3. $-2$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

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

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

  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 } } }

    $
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

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:

  1. 5

  2. -5

  3. 4

  4. -4
Reveal answer Fill a bubble to check yourself
A,D Correct answer
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 $a = 5, -4$

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

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

  1. 5

  2. -5

  3. 4

  4. -4
Reveal answer Fill a bubble to check yourself
A,D Correct answer
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 $a = 5, -4$

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

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

  1. two values of $a$
  2. for all $a$
  3. for one value of $a$
  4. for no values of $a$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $\displaystyle D> 0$

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

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

  1. equilateral

  2. isosceles

  3. right angled triangle

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
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 .