JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Straight Lines

Let the equation of the pair of lines,

y = p x

and

y = q x

, can be written as

(y - p x) (y - q x) = 0

. Then the equation of the pair of the angle bisectors of the lines

x^{2} - 4 x y - 5 y^{2} = 0

is :

(A) $x^{2} - 3 x y + y^{2} = 0$
(B) $x^{2} + 3 x y - y^{2} = 0$ ✓ Correct
(C) $x^{2} - 3 x y - y^{2} = 0$
(D) $x^{2} + 4 x y - y^{2} = 0$

Answer: (B)

Solution

For a pair of lines

a x^{2} + 2 h x y + b y^{2} = 0

the angle bisectors satisfy

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

. Here

a = 1

2 h = - 4

h = - 2

, and

b = - 5

, giving

\frac{x ^{2} - y ^{2}}{6} = \frac{x y}{- 2}

that is

- 2 (x^{2} - y^{2}) = 6 x y

. Dividing by

- 2

and rearranging, the pair of angle bisectors is

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

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Single correct option

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