Q107 / 160 · TS EAMCET 2020 (11 Sep Shift 1) · Mathematics
If the tangent and normal drawn to the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$ at $P\left(\theta=\frac{\pi}{2}\right)$ cuts the $\mathrm{X}$-axis at $A$ and $B$ respectively, then the area (in sq. units) of $\triangle P A B$ is
- A$\frac{a^2}{\sqrt{2}}$
- B$\frac{\sqrt{2}}{a^2}$
- Correct answerC$a^2$
- D$2 a^2$
Solution
Given curve is
$x=a(\theta+\sin \theta), y=a(1-\cos \theta)$
$\therefore \quad \frac{d y}{d x}=\frac{\sin \theta}{1+\cos \theta}=\tan \frac{\theta}{2}$
$\therefore$ Slope of tangent and normal drawn to the given curve at $P\left(\theta=\frac{\pi}{2}\right)$ is $m_T=1$ and $m_N=-1$ respectively.
Now, equation of tangent to the given curve at point $P\left(a\left(\frac{\pi}{2}+1\right), a\right)$ is $y-a=1[x-a(\pi / 2+1)]$
$\therefore$ Point $A$ is $\left(\frac{a \pi}{2}, 0\right)$
and equation of normal to the given curve at point $P$ is $y-a=-1\left[x-a\left(\frac{\pi}{2}+1\right)\right]$
So, point $B$ is $\left(a\left(\frac{\pi}{2}+2\right), 0\right)$
Therefore area of $\triangle P A B=\frac{1}{2}\left|\begin{array}{ccc}a\left(\frac{\pi}{2}+1\right) & a & 1 \\ \frac{a \pi}{2} & 0 & 1 \\ a\left(\frac{\pi}{2}+2\right) & 0 & 1\end{array}\right|$
$=\frac{1}{2}\left|a\left[\frac{a \pi}{2}-\frac{a \pi}{2}-2 a\right]\right|=a^2$
Hence, the correct answer is option: 'C'.