Question

Find the equation to the tangent and normal at the point of the ellipse $5x^2+3y^2=137$ whose ordinate is $2$.

Answer 1

$5x^2+{3y}^2=137$

Ordinate $y=2$

$5x^2+12=1375x^2=125\Rightarrow x=5$

Point on the ellipse is $(5,2)$

For finding the equation of tangent at any point $(x^{\prime },y^{\prime })$ on ellipse, we replace $x$ by $x{x}^{\prime }$ and $y$ by $y{y}^{\prime }$

So the equation of tangent at $(5,2)$ is

$5\left(5x\right)+3\left(2y\right)=13725x+6y=137$

Slope $m=\frac{-25}{6}$

Let slope of normal at this point be $m^{\prime }$

$m{m}^{\prime }=-1\frac{-25}{6}{m}^{\prime }=-1\Rightarrow m^{\prime }=\frac{6}{25}$

Equation of normal

$y-2=\frac{6}{25}(x-5)25y-50=6x-306x-25y+20=0$