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?