Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#6

Question

The diagram shows a circle with radius r cm and centre O. The line PT is the tangent to the circle at  P and angle  radians. The line OT meets the circle at Q.

i.       Express the perimeter of the shaded region PQT in terms of r and .

ii.       In the case where  and , find the area of the shaded region correct to 2 significant  figures.

Solution

i.

It is evident from the diagram that;

Let’s first of all find arc PQ.

Expression for length of a circular arc with radius  and angle  rad is;

For arc PQ;

Now we find PT.

Tangent at any point of the circle is perpendicular to the radius through the point of contact.

Consider . It is evident that  is a right angle.

Expression for  trigonometric ratio in right-triangle is;

For right angled triangle OPT;

Next, we find QT.

It is evident from the diagram that;

To find OT, consider right angled triangle OPT.

Pythagorean Theorem

For right angled triangle OPT;

We know that  and from above  we have .

Therefore;

Finally;

ii.

It is evident from the diagram that;

Let’s first of all find area of triangle OPT.

Expression for the area of the triangle is;

For arc triangle OPT;

We know that  and from above  we have . Therefore;

Next we find area of sector OPQ.

Expression for area of a circular sector with radius  and angle  rad is;

For sector OPQ;

Hence, we can find area of shaded region;

We are given that  and ;