Those are just the tangents to each curve, at the points of intersection.

A **common tangent** is when the tangents to both curves (at their points of intersection) are **the same line**. The curves are essentially tangential to each other when this appears.

Nah, I think that diagram is showing lines that happen to be tangents of both the circle and parabola, but not at the same point. It's not what I'd define as a

**common** tangent either but it makes the question answerable at least!

As to finding it - I'm honestly not sure! On first thought, you need a point on the parabola \(x_1,y_1\), and a point on the parabola \(x_2,y_2\), such that the gradient of the interval joining the two points is equal to the gradient of the curve at both points. To me that's two equations with four variables. You could then use the equations of the curves (for example, \(y_1=x_1(x_1-4)\)) to get rid of two of the variables, leaving two equations in two variables - Simulteneously solve

(I anticipate that algebra to be

**horrendous**)