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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 5 - 2022 - Paper 1

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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 5-2022-Paper 1.png

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 ... 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 5 - 2022 - Paper 1

Step 1

Find the coordinates of the centre of the circle

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Answer

To find the centre of the circle, we need to rewrite the equation in the standard form of a circle, which is

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

First, we complete the square for the x and y terms in the equation:

  1. Rearranging the equation: x2−10x+y2+16y=80x^2 - 10x + y^2 + 16y = 80

  2. Completing the square for xx: x2−10x=(x−5)2−25x^2 - 10x = (x - 5)^2 - 25

  3. Completing the square for yy: y2+16y=(y+8)2−64y^2 + 16y = (y + 8)^2 - 64

  4. Substituting these into the equation gives: (x−5)2−25+(y+8)2−64=80(x - 5)^2 - 25 + (y + 8)^2 - 64 = 80

  5. Simplifying, we have: (x−5)2+(y+8)2=169(x - 5)^2 + (y + 8)^2 = 169

The coordinates of the centre are therefore (5,−8)(5, -8).

Step 2

Find the radius of the circle

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Answer

From the standard form of the circle, we see that:

r2=169r^2 = 169

Thus, the radius is: r=extsqrt(169)=13r = ext{sqrt}(169) = 13.

Step 3

Find the exact length OP

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Answer

To find the length OPOP, where point PP is the point on the circle that is furthest from the origin OO, we can use the coordinates of the centre (5,−8)(5, -8) and the radius 1313.

The distance from the origin to the centre is: d(O,C)=extsqrt(52+(−8)2)=extsqrt(25+64)=extsqrt(89)d(O, C) = ext{sqrt}(5^2 + (-8)^2) = ext{sqrt}(25 + 64) = ext{sqrt}(89)

To find the length OPOP, we add the radius to this distance: OP=d(O,C)+r=extsqrt(89)+13OP = d(O, C) + r = ext{sqrt}(89) + 13

Thus, the exact length OPOP is: OP = rac{ ext{sqrt}(89) + 13}{1}.

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