Question

The angle between the lines represented by the equation $4x^2-24xy+11y^2=0$ are

• A. ${\tan }^{-1}\left(\frac{3}{4}\right)$
• B. ${\tan }^{-1}\left(\frac{1}{3}\right)$
• C. ${\tan }^{-1}\left(\frac{4}{3}\right)$
• D. ${\tan }^{-1}\left(\frac{1}{2}\right)$

Answer 1

The correct option is C

${\tan }^{-1}\left(\frac{4}{3}\right)$

Explanation for correct option:

Finding the angle between the lines:

Given the equation $4x^2-24xy+11y^2=0$

Comparing the given equation with $a{x}^2-2hxy+b{y}^2=0$ we get,

$a=4$,$b=11$,and $2h=-24\Rightarrow h=-12$

Finding the angle between the line using the formula,

$\begin{array}{l}\tan \theta =\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|\\ =\left|\frac{2\sqrt{(-12{)}^2-(4\left)\right(11)}}{4+11}\right|\\ =\left|\frac{2\sqrt{144-44}}{15}\right|\\ =\left|\frac{2\sqrt{100}}{15}\right|\\ =\left|\frac{2\times 10}{15}\right|\\ =\left|\frac{4}{3}\right|\end{array}$

Therefore, $\tan \left(\theta \right)=\pm \frac{4}{3}$ after removing the modulus sign.

The positive value of $\tan \left(\theta \right)=\frac{4}{3}$

Therefore the angles between the lines are represented by the equation $4x^2-24xy+11y^2=0$ are $\theta ={\tan }^{-1}\left(\frac{4}{3}\right)$

Hence, option (C) is the correct answer