Review question

# Where does this tangent to this ellipse meet $x=a$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9325

## Suggestion

The point $P$ in the first quadrant lies on the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. The points $A^{\prime} \ (-a, 0)$ and $A \ (a, 0)$ are the extremities of the major axis of the ellipse, and $O$ is the origin. The tangent to the ellipse at $P$ meets the axis of $y$ at $Q$ and meets the line $x = a$ at $T$. The chord $A^{\prime} P$ meets the axis of $y$ at $M$, and when produced meets the line $x=a$ at $R$.

Prove that

1. $AT=TR$,

2. $OQ^2 - MQ^2 = b^2$.

How can we find the tangent to the ellipse? Be careful when you differentiate the equation of the ellipse!

Do not forget that we know that the point $P$ lies on the ellipse.