The combined equation of the two lines a x + b y + c = 0 and a ' x + b ' y + c ' = 0 can be written as ( a x + b y + c ) ( a ' x + b ' y + c ' ) = 0 . The equation of the angle bisectors of the lines represented by the equation 2 x 2 + x y

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Q.
The combined equation of the two lines $a x + b y + c = 0$ and $a' x + b' y + c' = 0$ can be written as $(a x + b y + c) (a' x + b' y + c') = 0$ . The equation of the angle bisectors of the lines represented by the equation $2 x^{2} + x y - 3 y^{2} = 0$ is [2023]

1 $3 x^{2} + 5 x y + 2 y^{2} = 0$

2 $x^{2} - y^{2} - 10 x y = 0$

3 $3 x^{2} + x y - 2 y^{2} = 0$

4 $x^{2} - y^{2} + 10 x y = 0$

Ans.
(2)

$Given equation of lines is 2 x^{2} + x y - 3 y^{2} = 0$

$Equation of angle bisector of lines a x^{2} + 2 h x y + b y^{2} = 0 is$

$\frac{x^{2} - y^{2}}{x y} = \frac{a - b}{h}$

$Here, a = 2, b = - 3$ and $h = \frac{1}{2}$

$∴$ Equation of angle bisector of given lines is

$\frac{x^{2} - y^{2}}{x y} = \frac{2 + 3}{\frac{1}{2}} = 10$

$\Rightarrow x^{2} - y^{2} = 10 x y i.e., x^{2} - y^{2} - 10 x y = 0$

Solution

Q.
The combined equation of the two lines $a x + b y + c = 0$ and $a' x + b' y + c' = 0$ can be written as $(a x + b y + c) (a' x + b' y + c') = 0$ . The equation of the angle bisectors of the lines represented by the equation $2 x^{2} + x y - 3 y^{2} = 0$ is [2023]

1 $3 x^{2} + 5 x y + 2 y^{2} = 0$

2 $x^{2} - y^{2} - 10 x y = 0$

3 $3 x^{2} + x y - 2 y^{2} = 0$

4 $x^{2} - y^{2} + 10 x y = 0$

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Solution

Q. The combined equation of the two lines ax+by+c=0 and a'x+b'y+c'=0 can be written as (ax+by+c)(a'x+b'y+c')=0. The equation of the angle bisectors of the lines represented by the equation 2x2+xy-3y2=0 is [2023]

1 3x2+5xy+2y2=0

2 x2-y2-10xy=0

3 3x2+xy-2y2=0

4 x2-y2+10xy=0

Ans. (2) Given equation of lines is 2x2+xy-3y2=0 Equation of angle bisector of lines ax2+2hxy+by2=0 is x2-y2xy=a-bh Here, a=2, b=-3 and h=12 ∴ Equation of angle bisector of given lines is x2-y2xy=2+312=10 ⇒ x2-y2=10xy i.e., x2-y2-10xy=0

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Q.
The combined equation of the two lines $a x + b y + c = 0$ and $a' x + b' y + c' = 0$ can be written as $(a x + b y + c) (a' x + b' y + c') = 0$ . The equation of the angle bisectors of the lines represented by the equation $2 x^{2} + x y - 3 y^{2} = 0$ is [2023]

1 $3 x^{2} + 5 x y + 2 y^{2} = 0$

2 $x^{2} - y^{2} - 10 x y = 0$

3 $3 x^{2} + x y - 2 y^{2} = 0$

4 $x^{2} - y^{2} + 10 x y = 0$