The tangent and normal to the ellipse
Question:

The tangent and normal to the ellipse $3 x^{2}+5 y^{2}=32$ at the point $P(2,2)$ meet the $x$-axis at $Q$ and $R$, respectively. Then the area (in sq. units) of the triangle $P Q R$ is :

  1. $\frac{14}{3}$

  2. $\frac{16}{3}$

  3. $\frac{68}{15}$

  4. $\frac{34}{15}$


Correct Option: , 3

Solution:

$3 x^{2}+5 y^{2}=32$

$\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{(2,2)}=-\frac{3}{5}$

Tangent : $\mathrm{y}-2=-\frac{3}{5}(\mathrm{x}-2) \Rightarrow \mathrm{Q}\left(\frac{16}{3}, 0\right)$

Normal : $\mathrm{y}-2=\frac{5}{3}(\mathrm{x}-2) \Rightarrow \mathrm{R}\left(\frac{4}{5}, 0\right)$

Area is $=\frac{1}{2}(\mathrm{QR}) \times 2=\mathrm{QR}=\frac{68}{15}$

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