A circle has equation $x^2 + y^2 - 6x - 8y = 264$ - AQA - A-Level Maths Pure - Question 5 - 2019 - Paper 3
Question 5
A circle has equation $x^2 + y^2 - 6x - 8y = 264$.
AB is a chord of the circle.
The angle at the centre of the circle, subtended by AB, is 0.9 radians, as shown ... show full transcript
Worked Solution & Example Answer:A circle has equation $x^2 + y^2 - 6x - 8y = 264$ - AQA - A-Level Maths Pure - Question 5 - 2019 - Paper 3
Step 1
Find radius of the circle
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Answer
To find the radius of the circle, we start with the equation:
x2+y2−6x−8y=264
We can complete the square for the x and y terms:
For x2−6x, we add and subtract (6/2)2=9.
For y2−8y, we add and subtract (8/2)2=16.
Thus, we rewrite the equation as:
(x−3)2+(y−4)2=289
This shows us that the center of the circle is at (3, 4) and the radius is:
r=extsqrt(289)=17.
Step 2
Find area of the sector
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Answer
The area of a sector (A) is given by:
A=21r2θ
Substituting the radius (17) and angle (0.9 radians):
A=21×172×0.9=130.05.
Step 3
Find area of triangle
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Answer
The area of triangle formed by the radius and chord (A_triangle) can be calculated using:
Atriangle=21r2sin(θ)
Substituting the radius (17) and angle (0.9 radians):
Atriangle=21×172×sin(0.9)≈113.19.
Step 4
Find area of the minor segment
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Answer
The area of the minor segment (A_segment) is then given by:
Asegment=Asector−Atriangle
Substituting the values calculated:
Asegment=130.05−113.19=16.86
Rounded to three significant figures, the final answer is 16.9.
