# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#8

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Question

In the diagram, AB is an arc of a circle with centre O and radius . The line XB is a tangent to the circle at B and A is the mid-point of OX.

Express each of the following in terms of,  and :

ii. the perimeter of the shaded region,

iii. the area of the shaded region.

Solution

i.

We are given that BX is tangent to the curve at point B, therefore;

Now consider the .

Expression for  trigonometric ratio in right-triangle is;

Hence;

We are given that  and since A is the mid-point of  OX and , therefore;

Hence;

ii.

It is evident from the diagram that;

We are given that  and since A is the mid-point of  OX and , therefore;

Now to find  we consider the .

Expression for  trigonometric ratio in right-triangle is;

For the ;

From (i), we have;

We are also given that . Therefore;

Now we need length of arc AB.

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

For the given case;

Hence;

iii.

It is evident from the diagram that;

First we find area of .

Expression for the area of the triangle is;

For ;

We are given that  and from (i) we have . Therefore;

Now we find area of sector OAB.

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

For sector OAB.

We are given that radius  and from (i), we have .

Therefore;

Hence;