Review question

# What can we say if a normal to an ellipse passes through this point? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5237

## Suggestion

Prove that the equation of the normal to the ellipse $x^2/a^2+y^2/b^2=1$ at the point $P\,(a\cos\theta,b\sin\theta)$ is $\frac{ax}{\cos \theta} - \frac{by}{\sin \theta} = a^2-b^2.$

If $x = f(t), y = g(t)$, what is $\dfrac{dy}{dx}$ in terms of $t$?

Or could we use implicit differentiation?

If there is a value of $\theta$ between $0$ and $\pi/2$ such that the normal at $P$ passes through one end of the minor axis, show that the eccentricity of the ellipse must be greater than $1/\sqrt{2}$.

You might use this applet to explore the problem.