Review question

# Where do two tangents cross, and where do two normals cross? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5703

## Question

The curve $C$ has equation $xy=\frac{1}{2}$. The tangents to the curve $C$ at the distinct points $P(p,\frac{1}{2p})$ and $Q(q,\frac{1}{2q})$, where $p$ and $q$ are positive, intersect at $T$ and the normals to $C$ at these points intersect at $N$. Show that $T$ is the point $\left(\frac{2pq}{p+q},\frac{1}{p+q}\right).$ In the case $pq=\frac{1}{2}$, find the coordinates of $N$. Show (in this case) that $T$ and $N$ lie on the line $y=x$ and are such that the product of their distances from the origin is constant.