# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/11) | Q#6

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Question

The diagram shows a circle with centre O and radius r cm. The points A and B lie on  the circle and AT is a tangent to the circle. Angle radians and OBT is a  straight line.

i.       Express the area of the shaded region in terms of r and .

ii.       In the case where  and , find the perimeter of the shaded region.

Solution

i.

It is evident from the diagram that;

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

Expression for the area of the triangle is;

For arc triangle OAT;

We know that .

Let’s we find AT.

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 since AT is a tangent to the  circle.

Expression for  trigonometric ratio in right-triangle is;

For right angled triangle OAT;

Hence;

Next we find area of sector AOB.

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

For sector AOB;

Hence, we can find area of shaded region;

ii.

It is evident from the diagram that;

Let’s first of all find length of arc AB.

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

For arc AB;

For AT, we have demonstrated in (i);

Now we find BT.

It is evident from the diagram that;

To find OT, consider right angled triangle OAT.

Pythagorean Theorem

For right angled triangle OAT;

We know that and from above  we have .

Therefore;

Finally;

We are given that  and ;