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

Correct answer: D. x ^ 2 - y ^ 2 - 10xy = 0 Equation of the pair of angle bisectors for the homogenous equation a x ^ 2 + 2hxy + b y ^ 2 = 0 is given as x ^ 2 - y ^ 2 a - b = (xy)/(h) Here, a = 2, h = (1)/(2) and b = - 3 Equation will become: x ^ 2 - y ^ 2 2 - (- 3) = (xy)/((1)/(2)) ⇒ x ^ 2 - y ^ 2 = 10xy ⇒ x ^ 2 - y ^ 2 - 10xy = 0

JEE Main 20231 FebruaryMorning Shift

Exam: JEE Main
Year: 2023
Date: 1 February
Shift: Morning Shift
Subject: Mathematics
Chapter: Straight Lines

JEE Main 2023 · 1 February · Morning Shift · Mathematics · Straight Lines

The combined equation of the two lines ax + by + c = 0 and a ^ ' x...

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

Options

A $3{x}^{2} + 5xy + 2{y}^{2} = 0$
B ${x}^{2} - {y}^{2} + 10xy = 0$
C $3{x}^{2} + xy - 2{y}^{2} = 0$
D ${x}^{2} - {y}^{2} - 10xy = 0$

Correct answer

D. ${x}^{2} - {y}^{2} - 10xy = 0$

Step-by-step solution

Equation of the pair of angle bisectors for the homogenous equation $a{x}^{2} + 2hxy + b{y}^{2} = 0$ is given as

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

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

Equation will become:

$\dfrac{{x}^{2} - {y}^{2}}{2 - \left(- 3\right)} = \dfrac{xy}{\dfrac{1}{2}}$

$⇒ {x}^{2} - {y}^{2} = 10xy$

$⇒ {x}^{2} - {y}^{2} - 10xy = 0$

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