A circle C has centre M (6, 4) and radius 3 - Edexcel - A-Level Maths Pure - Question 9 - 2008 - Paper 2

To find the angle TMQ, we can use the properties of the tangent and the radii of the circle. The coordinates of T are on the circle, specifically, we can calculate the slope of the tangent line at that point and the slope of the line from M to Q.

The angle can be determined using the tangent function:

$\tan(\theta) = \frac{\text{slope of TM}}{\text{slope of MQ}}.$
Solving for the angle using the inverse tangent function yields the desired angle TMQ to be approximately 1.0766 radians.

To find the area of the shaded region TPQ, we will calculate the area of triangle TPQ and then subtract the area under the curve TQ.

1. The area of triangle TPQ can be determined using:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}.$
We can determine the base and height using the coordinates of points P and Q.

2. The area under the curve can be found using integration methods over the limits defined by the points T to P to Q.

By calculating these areas and then simplifying, we can express the area of TPQ. Finally, we report the area to three decimal places.