Review question

# Can we find the areas enclosed between a circle and its tangent? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8946

## Question

In the diagram below is sketched the circle with centre $(1,1)$ and radius $1$ and a line $L$. The line $L$ is tangential to the circle at $Q$. Further, $L$ meets the $y$-axis at $R$, and the $x$-axis at $P$ in such a way that the angle $OPQ$ equals $\theta$ where $0<\theta<\dfrac{\pi}{2}$.

1. Show that the co-ordinates of $Q$ are

$(1+ \sin \theta, 1 + \cos \theta),$

and that the gradient of $PQR$ is $- \tan \theta$.

Write down the equation of the line $PQR$ and so find the co-ordinates of $P$.

2. The region bounded by the circle, the $x$-axis and $PQ$ has area $A(\theta)$; the region bounded by the circle, the $y$-axis and $QR$ has area $B(\theta)$. (See diagram.)

Explain why

$A(\theta) = B\left(\dfrac{\pi}{2} - \theta\right)$

for any $\theta$.

Calculate $A\left(\dfrac{\pi}{4}\right)$.

3. Show that

$A \left(\dfrac{\pi}{3} \right) = \sqrt{3} - \dfrac{\pi}{3}.$