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}

 

Answers (1)
V Vakul

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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