Review question

# Can we find the triangle areas given by a tangent? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8286

## Suggestion

Find the equation of the tangent to the rectangular hyperbola $xy=c^2$ at the point $P(ct,c/t)$.

A well drawn diagram will help us a lot in this question!

What is the gradient at any point on the curve $xy=c^2$?

The tangent at $P$ meets the axes of $x$ and $y$ at $L$ and $M$ respectively. $O$ is the centre of the hyperbola and $POQ$ is a diameter. The straight line $MQ$ meets the axis of $x$ in $T$. Prove that

1. the area of triangle $MOL$ is $2c^2$,
2. the area of triangle $QOT$ is $c^2/3$.

Can we use the equation of the tangent at $P$ to find $L$ and $M$?

What are the coordinates of $Q$ if $POQ$ is a diameter?

Note that a diameter of a hyperbola is any straight line passing through its centre. We can assume here that $Q$ is the point on the hyperbola opposite $P$.