Review question

# Can we show these tangents to this hyperbola form a square? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7833

## Suggestion

Sketch on the same axes the two curves $C_1$ and $C_2$, given by \begin{align*} C_1&: xy &=1, \\ C_2&: x^2-y^2&=2. \end{align*}

Can we sketch the hyperbola $xy=1$?

Can we sketch the hyperbola $x^2-y^2=2$?

The curves intersect at $P$ and $Q$. Given that the coordinates of $P$ are $(a,b)$ (which you need not evaluate), write down the coordinates of $Q$ in terms of $a$ and $b$.

Does our sketch tell us what the coordinates of $Q$ are?

The tangent to $C_1$ through $P$ meets the tangent to $C_2$ through $Q$ at the point $M$, and the tangent to $C_2$ through $P$ meets the tangent to $C_1$ through $Q$ at $N$. Show that the coordinates of $M$ are $(-b,a)$ and write down the coordinates of $N$.

Could we find the equations of the tangents and then try to solve them together?

Along the way, we could need to use the information that $(a,b)$ lies on both $C_1$ and $C_2$. That is, we know that $ab = 1$ and $a^2 - b^2 = 2$.

Show that $PMQN$ is a square.

What will the gradients of the tangents have to multiply to?

What must be true for the sides of the quadrilateral?