Question

If the equation $a{x}^2+2hxy+b{y}^2=0$ has the one line as the bisector of angle between the coordinates axes, then

• A. ${(a–b)}^2=h^2$
• B. ${(a+b)}^2=h^2$
• C. ${(a–b)}^2=4h^2$
• D. ${(a+b)}^2=4h^2$

Answer 1

The correct option is D

${(a+b)}^2=4h^2$

Explanation for correct option

Let the auxiliary equation of $a{x}^2+2hxy+b{y}^2=0$ is $b{m}^2+2hm+a=0$

It is given that one of the line bisects the angle between the co-ordinate axes.

So the line makes an angle $45°$ or $135°$ with the positive direction of $x$-axis.

$\therefore m=\tan \left(45°\right)=1$

or $m=\tan \left(135°\right)=-1$

Therefore $m=\pm 1$ are the roots of the auxiliary equation $b{m}^2+2hm+a=0$

$$\begin{gathered} \therefore b{\left(\pm 1\right)}^2+2h\left(\pm 1\right)+a=0 \\ \Rightarrow b+a=\pm 2h \\ \Rightarrow {\left(b+a\right)}^2=4h^2 \end{gathered}$$

Hence, Option (D) is correct i.e. ${(a+b)}^2=4h^2$