Question

The lines $2x^2+6xy+y^2=0$ are equally inclined to the lines $4x^2+18xy+b{y}^2=0$ when $b=1$

• A. True
• B. False

Answer 1

The correct option is A True

Consider the equation $2x^2+6xy+y^2=0$

Equation of angle bisector is

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

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

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

Consider the equation $4x^2+18xy+b{y}^2=0$

Equation of angle bisector is

$\frac{x^2-y^2}{4-b}=\frac{xy}{9}$

$\frac{x^2-y^2}{xy}=\frac{4-b}{9}$ .......$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$ we have

$\frac{x^2-y^2}{xy}=\frac{1}{3}=\frac{4-b}{9}$

$\Rightarrow \frac{1}{3}=\frac{4-b}{9}$

$\Rightarrow 4-b=\frac{9}{3}=3$

$\Rightarrow -b=3-4$

$\Rightarrow b=1$

$\therefore b=1$