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Equation of a Circle0%

Circles · Topic 2 of 7

Equation of a Circle

Video lesson · from 0:191 worked example

One lesson video covers all of Circles, so it opens at 0:19 for this topic — not from the beginning.

Theory

The equation of a circle with centre (a,b)(a,b) and radius rr units is:

(x−a)2+(y−b)2=r2(x-a)^2+(y-b)^2=r^2
xy(a, b)(x, y)ry - bx - a

Task: Complete the tables

Equation of CircleCentreRadius
(x−5)2+(y−1)2=82(x-5)^2+(y-1)^2=8^2(5,1)(5, 1)8
(x+5)2+(y+1)2=100(x+5)^2+(y+1)^2=100(−5,−1)(-5, -1)10
(x−2)2+(y+1)2=65(x-2)^2+(y+1)^2=65(2,−1)(2, -1)65\sqrt{65}
CentreRadiusEquation of Circle
(4,9)(4, 9)7 units(x−4)2+(y−9)2=49(x-4)^2+(y-9)^2=49
(−2,9)(-2, 9)3\sqrt{3} units(x+2)2+(y−9)2=3(x+2)^2+(y-9)^2=3
(−2,−6)(-2, -6)252\sqrt{5} units(x+2)2+(y+6)2=20(x+2)^2+(y+6)^2=20

⚠️ Common Examiner Traps

  • The signs inside the brackets flip: a centre of (3,−2)(3,-2) gives (x−3)2+(y+2)2=r2(x-3)^2 + (y+2)^2 = r^2. Reading the centre straight out of the brackets without changing sign is the classic slip.
  • It is r2r^2 on the right: so a radius of 5 gives 25. Going the other way, remember to square root before quoting the radius.
  • Find the radius from two points if needed: the distance from the centre to a point on the circle is the radius — use the distance formula and keep it exact.
  • Do not expand unless asked: the centre-radius form is usually the most useful one to leave the answer in.

Worked examples

Example 1

The circle with centre (1,4)(1,4) also passes through the point (5,6)(5,6). Find the equation of this circle.

The equation is of the form (x−a)2+(y−b)2=r2(x-a)^2+(y-b)^2=r^2.

Centre (a,b)=(1,4)(a,b) = (1,4), so:

(x−1)2+(y−4)2=r2(x-1)^2+(y-4)^2=r^2

Substitute the point (5,6)(5,6) to find r2r^2:

(5−1)2+(6−4)2=r2(5-1)^2+(6-4)^2=r^2
(4)2+(2)2=r2(4)^2+(2)^2=r^2
16+4=r216+4=r^2
r2=20r^2=20

So the equation is:

(x−1)2+(y−4)2=20(x-1)^2+(y-4)^2=20