Question

If the sum of the slope of the line represented by the equation ${x^2} - 2xy\tan A - {y^2} = 0$ be $4$ then $\angle A = $
A. $0$
B. ${45^ \circ }$
C. ${60^ \circ }$
D. ${\tan ^{ - 1}}\left( { - 2} \right)$

Answer 1

Hint: First we will compare the given equation ${x^2} - 2xy\tan A - {y^2} = 0$ with the general equation $a{x^2} + 2hxy + b{y^2} = 0$ and find the value of $a$, $b$ and $h$. Then put these values in the formula for sum of slopes. The after solving we will get the required value of $\angle A$

Formula used: Sum of slopes$ = -\dfrac{{2h}}{b}$

Complete step by step solution: Given, equation is ${x^2} - 2xy\tan A - {y^2} = 0$
The general equation is $a{x^2} + 2hxy + b{y^2} = 0$
On Comparing with general equation, we get
$a = 1$, $h = - \tan A$ and $b = - 1$
Sum of slopes$ = 4$
Sum of slopes$ = -\dfrac{{2h}}{b}$
Putting values in above formula
$4 = - \dfrac{{ - 2\tan A}}{{ - 1}}$
After simplifying, we get
$4 = - 2\tan A$
$ - 2 = \tan A$
Taking ${\tan ^{ - 1}}$ on both sides
${\tan ^{ - 1}}\left( { - 2} \right) = {\tan ^{ - 1}}\left( {\tan A} \right)$
After solving, we get
$\angle A = {\tan ^{ - 1}}\left( { - 2} \right)$
Hence, the $\angle A$ is ${\tan ^{ - 1}}\left( { - 2} \right)$
Therefore, option d is correct
Additional Information:
The general form of equation is $a{x^2} + 2hxy + b{y^2} = 0$ and sum of slopes is $-\dfrac{2h}{b}$

Note: Students should solve the question step by step to avoid any calculation errors. They should know the required formula for solving the question. And take care of the negative sign in the formula.