Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/12) | Q#4

Question

The diagram shows a circle with centre O and radius cm. Points A and B lie on the circle and  angle radians. The tangents to the circle at A and B meet at T.

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

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

Solution


i.
 

We are required to find the perimeter of the shaded region.

It is evident from the diagram that;

First, we find length of arc AB.

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

Therefore, for the given case;

Now, we need to find AT & BT.

Consider the diagram below.

Tangent to a circle at a given point is perpendicular to the radius of that circle at that particular point.

Hence, OAT is right-angles triangle with angle radians and OBT is right-angles triangle with  angle  radians.

If two tangents to a circle meet at a point external to the circle, a line segments from the point of  intersection of tangents to the center of the circle bisects the angle of arc made between tangent  points on the circle.

Hence, OAT is right-angles triangle with angle radians and OBT is right-angles triangle  with angle  radians.

Expression for  trigonometric ratio in right-triangle is;

Therefore, for right-angles triangle OAT;

If two tangents to a circle meet at a point external to the circle, the line segments (tangents) from  respective tangent point to the point of intersection of tangents are equal in length.

Therefore, in the given case;

Hence;

Hence;


ii.
 

We are required to find the area of the shaded region.

It is evident from the diagram that;

First, we find area of triangle AOB.

Expression for the area of the triangle is;

Consider the diagram below.

As found in (i);

We are given that , therefore;

If two tangents to a circle meet at a point external to the circle, a line segments from the point of  intersection of tangents to the center of the circle bisects the angle of arc made between tangent  points on the circle and triangle AOB and triangle BOC are congruent right triangles.

Therefore, in the given case;

Next, we find area of sector AOB.

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

Finally, we can find area of shaded region.

We are given that and , therefore;

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