The absolute value of the tangent of the difference of the angles made by the lines $4 x^2-24 x y+11 y^2=0$…
The absolute value of the tangent of the difference of the angles made by the lines $4 x^2-24 x y+11 y^2=0$ with the $\mathrm{X}$-axis is
- $\frac{4}{11}$
- $\frac{24}{11}$
- $\frac{4}{3}$
- $\frac{11}{24}$
Solution
$\because 4 x^2-24 x y+11 y^2=0 \Rightarrow(2 x-y)(2 x-11 y)=0$
$\therefore$ Pair of equation of lines are
$2 x-y=0 \& 2 x-11 y=0$
and their angles from $x$-axis is:
$\begin{aligned}
& \theta_1=\tan ^{-1} 2 \& \theta_2=\tan ^{-1} \frac{2}{11} \\
& \text { Now, } \tan \left|\theta_1-\theta_2\right| \\
& =\tan \left|\tan ^{-1} 2-\tan ^{-1} \frac{2}{11}\right|=\tan \left|\tan ^{-1} \frac{2-\frac{2}{11}}{1+\frac{4}{11}}\right| \\
& =\tan \left(\tan ^{-1} \frac{4}{3}\right)=\frac{4}{3}
\end{aligned}$
Asked in: AP EAMCET 2023 (17 May Shift 1)
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