Question

If the sum of the slopes of the lines represented by the equation $x^2-2xy\tan A-y^2=0$ be $4$, then $\angle A=$

• A. $0°$
• B. $45°$
• C. $60°$
• D. ${\tan }^{-1}\left(-2\right)$

Answer 1

The correct option is D

${\tan }^{-1}\left(-2\right)$

Explanation for the correct option.

Let the slopes of the line $x^2-2xy\tan A-y^2=0$ be $m_1\mathrm{and}{m}_2$.

The general equation of pair of lines is $a{x}^2+2hxy+b{y}^2=0$.

By comparing the given equation with the general equation, we get $a=1,h=-\tan A,andb=-1$

Sum of slopes $=-\frac{2h}{b}$

$\begin{array}{rcl}& \Rightarrow & m_1+m_2=-\frac{2\tan A}{1}\\ \Rightarrow 4& =& -2\tan A\\ \Rightarrow \tan A& =& -2\\ \Rightarrow A& =& {\tan }^{-1}\left(-2\right)\end{array}$

Hence, option D is correct.