Question

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

• A. two pairs of perpendicular straight lines
• B. two pairs of parallel straight lines
• C. two pairs of straight lines which are equally inclined to each other
• D. none of these

Answer 1

The correct option is A two pairs of perpendicular straight lines

The pair of lines $a{x}^2+2h(a+b)xy+b{y}^2=0$ …………$\left(1\right)$

$a{x}^2+2hxy+b{y}^2=0$ …………..$\left(2\right)$

Equation for angle bisector for equation $\left(1\right)$ is

$\frac{x^2-y^2}{a^2-b^2}=\frac{xy}{h(a+b)}$

$\Rightarrow \frac{x^2-y^2}{a-b}=\frac{xy}{h}$ ……….$\left(3\right)$

Equation for angle bisector of equation $\left(2\right)$ is

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

Since $\left(3\right)$ & $\left(4\right)$ are equal

They are equally inclined to each other.