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10-2. Parabola, Ellipse, Hyperbola
hard
The tangent and normal to the ellipse $3x^2 + 5y^2 = 32$ at the point $P(2, 2)$ meet the $x-$ axis at $Q$ and $R,$ respectively. Then the area(in sq. units) of the triangle $PQR$ is
A
$\frac {34}{15}$
B
$\frac {68}{15}$
C
$\frac {14}{3}$
D
$\frac {16}{3}$
(JEE MAIN-2019)
Solution
$3{x^2} + 5{y^2} = 32$
${\left. {\frac{{dy}}{{dx}}} \right|_{\left( {2,2} \right)}} = – \frac{3}{5}$
tengent $:y – 2 = – \frac{3}{5}\left( {x – 2} \right) \Rightarrow Q\left( {\frac{{16}}{3},0} \right)$
Normal $:y – 2 = – \frac{5}{3}\left( {x – 2} \right) \Rightarrow R\left( {\frac{4}{5},0} \right)$
Area is $ = \frac{1}{2}\left( {QR} \right) \times 2 = \frac{{68}}{{15}}$
Standard 11
Mathematics
