Question

The lines represent by $a{x}^2+2hxy+b{y}^2=0$ are perpendicular to each other, if

• A. $h^2=a+b$
• B. $a+b=0$
• C. $h^2=ab$
• D. $h=0$

Answer 1

The correct option is B

$a+b=0$

Find the condition for the lines to be perpendicular

Given, $a{x}^2+2hxy+b{y}^2=0$

Let $\theta$ be the angle between the lines

Then, the formula for slope is

$\tan \left(\theta \right)=2\times \sqrt{\frac{h^2-ab}{a+b}}$

For lines to be perpendicular, $\theta =90°$

then, $\tan \left(90°\right)=2\times \sqrt{\frac{h^2-ab}{a+b}}$

$$\begin{gathered} \Rightarrow 2\times \sqrt{\frac{h^2-ab}{a+b}}=\infty \\ \Rightarrow a+b=0 \end{gathered}$$

Hence, the correct option is $B$, $a+b=0$.