The point \(P\) on the hyperbola \(xy=c^2\) is such that the tangent to the hyperbola at \(P\) passes through the focus of the parabola \(y^2=4ax\). Find the coordinates of \(P\) in terms of \(a\) and \(c\).

Every parabola has a special point, called the focus. If you imagine turning the parabola into a mirror, and turning it to face the sun, then all of the sun’s rays would be reflected by the parabola through its focus. For the parabola, \(y^2 = 4ax\), the focus is at \((a,0).\)

You might like to take a look at our Parabolic Mirrors resource.

What’s the equation of a line with gradient \(m\) that passes through \((a,0)\)? Where does this line intersect \(xy=c^2\)?

If the line is a tangent, how many times does it intersect the curve?

What does this tell us about \(m\)? Once we’ve found \(m\), can we find \(P\)?

If \(P\) also lies on the parabola, prove that \(a^4=2c^4\) …

The blue curve here is the hyperbola \(xy = c^2\), while the red curve is the parabola \(y^2=4ax\).

The green line is the tangent to the hyperbola though the parabola’s focus \(F\), which touches the hyperbola at \(P\).

The values of \(a^4\) and \(2c^4\) are given. Can we use this to verify what the question asks us to show?

… and calculate the acute angle between the tangents to the two curves at \(P\).

How is the gradient of a line linked to the angle it makes with the \(x\)-axis?

Or as an alternative method… the dot product of two vectors can help us find the angle between them.