Review question

# Where does this chord of an ellipse cut the $x$-axis? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6843

## Suggestion

Prove that the equation of the chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ joining the points $(a\cos\alpha,b\sin\alpha)$ and $(a\cos\beta,b\sin\beta)$ is $\frac{x}{a}\cos\left(\frac{\alpha+\beta}{2}\right)+\frac{y}{b}\sin\left(\frac{\alpha+\beta}{2}\right)= \cos\left(\frac{\alpha-\beta}{2}\right).$

What’s the equation of a line passing through two points whose coordinates we know?

Could we look up and use the standard formulae for $\sin x-\sin y$, and $\cos x-\cos y$?

Through a point $P$ on the major axis of an ellipse a chord $HK$ is drawn. Prove that the tangents at $H$ and $K$ meet the line through $P$ at right angles to the major axis at points equidistant from $P$.

Try varying $\alpha, \beta, a$ and $b$ here. The question asks us to show that $PH' = PK'$ (that the light green length equals the yellow length).

Does your experimenting back this up? Can we now turn this into a proof?

What are the coordinates of $P$?

How do we find the gradient of a tangent to an ellipse? What’s the equation of the tangent through $H$?

What are we trying to prove about the $y$-coordinates of $H'$ and $K'$?