# 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, The the area(in sq. units) of the triangle PQR is : Option 1) $\frac{34}{15}$ Option 2) $\frac{14}{3}$ Option 3) $\frac{16}{3}$ Option 4) $\frac{68}{15}$

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

$6x+10y\frac{dy}{dx}=0$

$=>\frac{dy}{dx}=\frac{-6x}{10y}$

$=>\frac{dy}{dx}=\frac{-3x}{5y}$

$m_T=\frac{-3}{5}$ => Equation of tangent at point (2,2)

$y-2=\frac{-3}{5}(x-2)$

$=>3x+5y=16$

$m_N=\frac{5}{3}$ => Equation of normal at point (2,2)

$y-2=\frac{5}{3}(x-2)$

$=>5x-3y=4$

$\therefore$ Area of $\Delta PQR$ = $\frac{1}{2}\times(\frac{16}{3}-\frac{4}{5})\times2=\frac{68}{15}$

So, option (4) is correct.

Option 1)

$\frac{34}{15}$

Option 2)

$\frac{14}{3}$

Option 3)

$\frac{16}{3}$

Option 4)

$\frac{68}{15}$

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