Review question

# When does a circle touch a parabola twice? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7979

## Question

The diagram below shows the parabola $y=x^2$ and a circle with centre $(0,2)$ just ‘resting’ on the parabola. By ‘resting’ we mean that the circle and parabola are tangential to each other at the points $A$ and $B$.

1. Let $(x,y)$ be a point on the parabola such that $x \ne 0$. Show that the gradient of the line joining this point to the centre of the circle is given by $\frac{x^2-2}{x}.$

2. With the help of the result from the first part, or otherwise, show that the coordinates of $B$ are given by $\left(\sqrt{\frac{3}{2}},\frac{3}{2} \right).$

3. Show that the area of the sector of the circle enclosed by the radius to $A$, the minor arc $AB$ and the radius to $B$ is equal to $\frac{7}{4} \cos^{-1} \left( \frac{1}{\sqrt{7}} \right) .$

4. Suppose now that a circle with centre $(0,a)$ is resting on the parabola, where $a > 0$. Find the range of values of $a$ for which the circle and parabola touch at two distinct points.

5. Let $r$ be the radius of a circle with centre $(0,a)$ that is resting on the parabola. Express $a$ as a function of $r$, distinguishing between the cases in which the circle is, and is not, in contact with the vertex of the parabola.