Tag: bisector of angle between lines

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Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

$P: 2x^{2}-axy+6y^{2}=0$
$Q: 3x^{2}-8xy+4y^{2}=0$
If the bisectors of the angles between the lines represented by $P$ and $Q$ are same, the value of $a$ is

  1. $8$
  2. $\dfrac{16}{3}$
  3. $32$
  4. $-16$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Equation of angle bisector of $2x^{ 2 }-axy+6y^{ 2 }=0$ is

$\cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ 2-6 } =\cfrac { xy }{ -a/2 } \Rightarrow a{ x }^{ 2 }-a{ y }^{ 2 }-8xy=0$   ----------(1)

And equation of angle bisector of $3x^{ 2 }-8xy+4y^{ 2 }=0$ is

$\cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ 3-4 } =\cfrac { xy }{ -4 } \Rightarrow 4{ x }^{ 2 }-4{ y }^{ 2 }-xy=0$  -----------(2)

From (1) and (2), we get $a=32$

Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

If the lines represented by $x^2-2pxy-y^2=0$ are rotated about the origin through an angle $\theta,$ one in clockwise direction and other in anti-clockwise direction, then the equation of the bisector of the angle between the lines in the new positions is

  1. $px^2+2xy-py^2=0$
  2. $px^2+2xy+py^2=0$
  3. $x^2-2pxy-y^2=0$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given eq 
$x^2-2pxy-y^2=0$
comparing above eq with general form of eq $ax^2+2hxy+by^2=0$
$a=1,h=-p,b=-1$
Now the line is rotated one in clockwise and other is anticlockwise so the both eq are replaced by each other and form the same eq as it was so here we finding eq of angle bisector by formula
$\dfrac{x^2-y^2}{a-b}=\dfrac{xy}{h}$
$\dfrac{x^2-y^2}{1+1}=\dfrac{xy}{-p}$
$\dfrac{x^2-y^2}{2}=\dfrac{xy}{-p}$
$(-p)(x^2-y^2)=2xy$
$-px^2+py^2=2xy$
$px^2+2xy-py^2=0$
 
Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines
$\displaystyle ax^{2}+2hxy+by^{2}=0$ represents a pair of straight lines through origin & angle between them is given by
$\displaystyle \tan \theta=\frac{2\sqrt{h^{2}-ab}}{a+b}$. If the lines are perpendicular then $\displaystyle a+b=0 $ and the equation of bisectors is given by  $\displaystyle \frac{x^{2}-y^{2}}{a-b}=\frac{xy}{h}$
The general equation of second degree given by
$\displaystyle ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$ represent a pair of straight lines if $\displaystyle \triangle =0 $ or 
$ \displaystyle \begin{vmatrix}a&h  &g \\ h&b  &f \\ g&f  &c \end{vmatrix}=0 $ or $\displaystyle abc+2fgh-af^{2}-bg^{2}-ch^{2}=0$
On the basis of above information answer the following question

The Joint equation of the bisectors of the angle between the lines represented by $\displaystyle ax^{2}+2hxy+by^{2}=0 $ is

  1. a pair of perpendicular lines

  2. a pair of parallel lines
  3. a pair of intersecting lines but not $\displaystyle \perp$er

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equation of bisectors of angle between the pair of straight line $\displaystyle ax^{2}+2hxy+by^{2}=0$ is given by $\displaystyle \frac{x^{2}-y^{2}}{a-b}=\frac{xy}{h}$


$\displaystyle\Rightarrow hx^{2}-\left ( a-b \right )xy-hy^{2}=0$

Here coefficient of $x^{2}+$ coefficient of $y^{2}=h+(-h)=0$

$\displaystyle \therefore $ above equation represent the pair of $\perp$er lines

Hence choice (a) is correct.

Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

The equation $a^2 x^2 + 2h(a+b) xy + b^2 y^2 = 0$ and $ax^2 + 2hxy + by^2 = 0$ represent

  1. two pairs of perpendicular straight lines

  2. two pairs of parallel straight lines

  3. two pairs of straight lines which are equally inclined to each other

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

$ax^2+2hxy+by^2=0$
Equation of the angle bisectors is given by
$\dfrac{x^2-y^2}{a-b}=\dfrac{xy}{h}$ ...(i)

For $ a^2x^2+2h(a+b)xy+b^2y^2=0$
The equations of the angle bisector is given by
$\dfrac{x^2-y^2}{a^2-b^2}=\dfrac{xy}{h(a+b)}$
 $\dfrac{x^2-y^2}{a-b}=\dfrac{xy}{h}$ ...(ii)
Since (i) is equal to (ii), the above lines are equally inclined to each other.

Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

If one of the lines of $my^2 + (1-m^2) xy - mx^2 = 0$ is a bisector of the angle between the lines $xy = 0$, then $m$ is

  1. $3$
  2. $2$
  3. $-\dfrac{1}{2}$
  4. $-1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$my^2+(1-m^2)xy-mx^2$
$(my+x)(y-mx)=0$
Therefore the lines are $y=mx$ and $y=\dfrac{-x}{m}$ ...(i)
The angle bisectors of the $xy=0$ is $y=x$ and $y=-x$
Comparing the slopes with the equations of the (i) we get
$m=\pm1$

Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

If one of the lines of is $my^{2}+\left ( 1-m^{2} \right )xy-mx^{2}=0$ is a bisector of the angle between the lines $\displaystyle xy = 0,$ then $m$ is

  1. $\displaystyle1$
  2. $\displaystyle2$
  3. $\displaystyle-\frac{1}{2}$
  4. $\displaystyle-1$
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation

Given pair of lines is
$\displaystyle { my }^{ 2 }+\left( 1-{ m }^{ 2 } \right) xy-{ mx }^{ 2 }=0\Rightarrow m{ \left( \frac { y }{ x }  \right)  }^{ 2 }+{ \left( 1-m \right)  }^{ 2 }\frac { y }{ x }- m=0$    ...(1)

Lines $xy=0$ are $x=0$ and $y=0.$
i.e the axes bisector of angle between the axes are $y=x$ and $y=-x$ $\displaystyle \Rightarrow \frac { y }{ x } =1$ or $\displaystyle \frac { y }{ x } =-1$

If $\displaystyle \frac { y }{ x } =1$ is represented by (1), then
$m{ \left( 1 \right)  }^{ 2 }+\left( 1-{ m }^{ 2 } \right) \left( -1 \right) -m=0\Rightarrow 1-{ m }^{ 2 }=0\Rightarrow m=\pm 1$

If $\displaystyle \frac { y }{ x } =-1$ is represented by (1), then
$m{ \left( -1 \right)  }^{ 2 }+\left( 1-{ m }^{ 2 } \right) \left( -1 \right) -m=0\Rightarrow 1-{ m }^{ 2 }=0\Rightarrow m=\pm 1$
In both cases either $m=1$ or $m=-1.$

Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

If the pair of straight lines ${x^2} - 2pxy - {y^2} = 0$ and ${x^2} - 2qxy - {y^2} = 0$ be such that each pair bisects the angle between the other pair,then:

  1. $pq=-1$
  2. $p=q$
  3. $p=-q$
  4. $pq=1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If each pair bisects the angle between the other, the lines must be the same, which implies the coefficients must match, leading to p = q.

Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

The pairs of straight lines $ax^{2}+2hxy-ay^{2}=0$ and $hx^{2}-2axy-hy^{2}=0$ are such that

  1. one pair bisects the angles between the other pair

  2. the lines of one pair are equally inclined to the lines of the other pair

  3. the lines of one pair are perpendicular to the `lines of the other pair

  4. none of these
Reveal answer Fill a bubble to check yourself
A,B Correct answer
Explanation

Given pairs of straight lines are $ax^{2}+2hxy-ay^{2}=0$ and $hx^{2}-2axy-hy^{2}=0$.

Pair of angular bisectors of $ax^{2}+2hxy-ay^{2}=0$ is $h(x^2-y^2)=(a-(-a))xy$

$\Rightarrow hx^2-2axy-hy^2$

$\therefore$ One pair bisects the angle between the other.

Clearly, if one pair bisects the angle between the other, the lines of one pair are equally inclined to the lines of the other pair.
Hence, option A and B.

Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

If one of the lines of $my^2 + (1- m^2) xy - mx^2 = 0$ is a bisector of the angle between the lines $xy = 0$, then $m$ is

  1. $1$
  2. $2$
  3. $\displaystyle \frac{-1}{2}$
  4. $-1$
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation

Angle bisectors of $xy=0$ are $x+y=0 $  and $x=y$
$my^2+xy-m^2 xy -mx^2=0$
$\therefore y(my+x)-mx(my+x)=0$
$\therefore (y-mx)(my+x)=0$
Comparing $y-mx=0  $ and $  my+x=0  $  with  $y=x $ and  $ y=-x$,  we get $m=\pm 1$ 

Multiple choice maths pair of straight lines bisection of angle pair of bisectors of angles bisector of angle between lines

The straight lines $7x^{2}+6xy+4y^{2}=0$ have the same pair of bisectors as those of the lines given by

  1. $49x^{2}+66xy+16y^{2}=0$
  2. $10x^{2}+6xy+7y^{2}=0$
  3. $5x^{2}+6xy+2y^{2}=0$
  4. $4x^{2}-6xy+7y^{2}=0$
Reveal answer Fill a bubble to check yourself
A,B,C,D Correct answer
Explanation

For $7x^{ 2 }+6xy+4y^{ 2 }=0$ 
Equation of angle bisector is 
$\cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ 7-4 } =\cfrac { xy }{ 3 } \Rightarrow { x }^{ 2 }-xy-{ y }^{ 2 }=0$
For Option A 
Equation of angle bisector of $49x^{ 2 }+66xy+16y^{ 2 }=0$ is
$\cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ 33 } =\cfrac { xy }{ 33 } \Rightarrow { x }^{ 2 }-xy-{ y }^{ 2 }=0$
For Option B
Equation of angle bisector of $10x^{ 2 }+6xy+7y^{ 2 }=0$ is
$\cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ 3 } =\cfrac { xy }{ 3 } \Rightarrow { x }^{ 2 }-xy-{ y }^{ 2 }=0$
For Option C
Equation of angle bisector of $5x^{ 2 }+6xy+2y^{ 2 }=0$ is
$\cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ 3 } =\cfrac { xy }{ 3 } \Rightarrow { x }^{ 2 }-xy-{ y }^{ 2 }=0$
For Option D
Equation of angle bisector of $4x^{ 2 }-6xy+7y^{ 2 }=0$ is
$\cfrac { { x }^{ 2 }-{ y }^{ 2 } }{ -3 } =\cfrac { xy }{ -3 } \Rightarrow { x }^{ 2 }-xy-{ y }^{ 2 }=0$