Question

The angle between the lines represented by the equation $\lambda x^2+{\left(1-\lambda \right)}^{2}xy-\lambda y^2=0$ is

• A. $30°$
• B. $45°$
• C. $60°$
• D. $90°$

Answer 1

The correct option is D

$90°$

Explanation for correct option:

Finding the angle between the lines:

Given equation is $\lambda x^2+{\left(1-\lambda \right)}^{2}xy-\lambda y^2=0$

Now comparing the given equation with the general form of an equation of a line $a{x}^2+2hxy+b{y}^2=0$

We get $a=\lambda$,$b=-\lambda$,$2h={\left(1-\lambda \right)}^2\Rightarrow h=\frac{{\left(1-\lambda \right)}^2}{2}$

Finding the angle between the two lines using the formula,

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

Substituting $a=\lambda$ ,$b=-\lambda$ and $h=\frac{{\left(1-\lambda \right)}^2}{2}$in the above equation, we get

$$\begin{gathered} \tan \theta =\left|\frac{2\sqrt{{\left({\displaystyle \frac{{\left(1-\lambda \right)}^2}{2}}\right)}^2-\left(\lambda \right)\left(-\lambda \right)}}{\lambda -\lambda }\right| \\ =\left|\frac{2\sqrt{\left({\displaystyle \frac{{\left(1-\lambda \right)}^4}{4}}\right)+{\lambda }^2}}{0}\right| \\ =\infty \end{gathered}$$

$$\begin{gathered} \tan \theta =\infty \\ \Rightarrow \theta ={\tan }^{-1}\left(\infty \right) \\ =90° \end{gathered}$$

Therefore, the angle between the lines represented by the equation is $90°$

Hence, option (D) is correct.