A circle has equation
$$x^2 + y^2 - 10x + 16y = 80$$
(a) Find
(i) the coordinates of the centre of the circle,
(ii) the radius of the circle - Edexcel - A-Level Maths Pure - Question 3 - 2022 - Paper 1
Question 3
A circle has equation
$$x^2 + y^2 - 10x + 16y = 80$$
(a) Find
(i) the coordinates of the centre of the circle,
(ii) the radius of the circle.
Given that P is th... show full transcript
Worked Solution & Example Answer:A circle has equation
$$x^2 + y^2 - 10x + 16y = 80$$
(a) Find
(i) the coordinates of the centre of the circle,
(ii) the radius of the circle - Edexcel - A-Level Maths Pure - Question 3 - 2022 - Paper 1
Step 1
Find (i) the coordinates of the centre of the circle
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Answer
To determine the center of the circle given by the equation x2+y2−10x+16y=80, we first rewrite it in standard form by completing the square for both the x and y terms.
Rearranging the equation:
x2−10x+y2+16y=80
Completing the square for the x terms:
x2−10x=(x−5)2−25
Completing the square for the y terms:
y2+16y=(y+8)2−64
Substitute these into the equation:
(x−5)2−25+(y+8)2−64=80
Simplifying gives:
(x−5)2+(y+8)2=169
Thus, the coordinates of the centre of the circle are at the point (5, -8).
Step 2
Find (ii) the radius of the circle
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Answer
From the standard form of the circle equation
(x−5)2+(y+8)2=169,
the radius can be found directly as the square root of the constant on the right side. Therefore, the radius ( r ) is:
r=169=13.
Hence, the radius of the circle is 13.
Step 3
Find (b) the exact length OP
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Answer
To find the point P on the circle that is furthest from the origin O(0, 0), we use the center of the circle (5, -8) and the radius, which is 13.
The distance from the origin to the center of the circle is given by:
d=(5−0)2+(−8−0)2=25+64=89
Since point P is the furthest point on the circle, we add the radius to this distance:
OP=89+13
