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A circle C has centre M (6, 4) and radius 3 - Edexcel - A-Level Maths Pure - Question 1 - 2008 - Paper 2
Question 1
A circle C has centre M (6, 4) and radius 3. (a) Write down the equation of the circle in the form $(x - a)^2 + (y - b)^2 = r^2$. (b) Show that the angle TMQ is 1... show full transcript
Worked Solution & Example Answer:A circle C has centre M (6, 4) and radius 3 - Edexcel - A-Level Maths Pure - Question 1 - 2008 - Paper 2
Step 1
Write down the equation of the circle in the form $(x - a)^2 + (y - b)^2 = r^2$
Answer
To write the equation of a circle in the specified form, we need the center coordinates (a, b) and the radius r.
Given that the center M is (6, 4) and the radius is 3, we can substitute these values into the formula:
Thus, the equation of the circle is:
Step 2
Show that the angle TMQ is 1.0766 radians to 4 decimal places
Answer
To find the angle TMQ, we can use the coordinates of points T, M, and Q. From the figure:
- The coordinates of T lie on the circle at a distance of 3 from M.
- We can find the coordinates of Q where the tangent at T meets the line MP.
Using the tangent properties and the dot product of vectors TM and MQ leads to the calculation of angle TMQ. Let's denote:
- as the radius from M to T
- as the line from M to Q
Calculate the angle using the formula:
heta = an^{-1} rac{opposite}{adjacent}After calculation, we find:
Step 3
Find the area of the shaded region TPQ. Give your answer to 3 decimal places.
Answer
To find the area of the shaded region TPQ, we will consider both the area of triangle TPQ and the area under the arc TQ.
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Area of Triangle TPQ: The vertices are T, P, and Q. The area can be calculated using the formula for the area of a triangle with vertices: ext{Area} = rac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2| Substitute the appropriate coordinates from T, P, and Q.
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Area under Arc TQ: This area can be calculated using the formula for the circular segment if the radius and angle are known.
After completing both calculations, sum the areas to get the total area of the shaded region TPQ. Present the answer rounded to 3 decimal places.