# 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

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 ;       