Ellipse 2 Question 2
2. The tangent and normal to the ellipse $3 x^{2}+5 y^{2}=32$ at the point $P(2,2)$ meets the $X$-axis at $Q$ and $R$, respectively. Then, the area (in sq units) of the $\triangle P Q R$ is
(a) $\frac{16}{3}$
(b) $\frac{14}{3}$
(c) $\frac{34}{15}$
(d) $\frac{68}{15}$
(2019 Main, 10 April II)
Show Answer
Solution:
- Equation of given ellipse is
$$ 3 x^{2}+5 y^{2}=32 $$
Now, the slope of tangent and normal at point $P(2,2)$ to the ellipse (i) are respectively
$$ m _T=\left.\frac{d y}{d x}\right| _{(2,2)} \text { and } m _N=-\left.\frac{d x}{d y}\right| _{(2,2)} $$
On differentiating ellipse (i), w.r.t. $x$, we get
$$ 6 x+10 y \frac{d y}{d x}=0 \Rightarrow \frac{d y}{d x}=-\frac{3 x}{5 y} $$
So, $m _T=-\left.\frac{3 x}{5 y}\right| _{(2,2)}=-\frac{3}{5}$ and $m _N=\left.\frac{5 y}{3 y}\right| _{(2,2)}=\frac{5}{3}$
Now, equation of tangent and normal to the given ellipse (i) at point $P(2,2)$ are
$$ (y-2)=-\frac{3}{5}(x-2) $$
and $(y-2)=\frac{5}{3}(x-2)$ respectively.
It is given that point of intersection of tangent and normal are $Q$ and $R$ at $X$-axis respectively.
So, $\quad Q \frac{16}{3}, 0$ and $R \frac{4}{5}, 0$
$\therefore$ Area of $\triangle P Q R=\frac{1}{2}(Q R) \times$ height
$$ \begin{gathered} =\frac{1}{2} \times \frac{68}{15} \times 2=\frac{68}{15} \text { sq units } \\ {\left[\because Q R=\sqrt{\frac{16}{3}-\frac{4}{5}^{2}}=\sqrt{\frac{68}{15}}=\frac{68}{15} \text { and height }=2\right]} \end{gathered} $$
