Circle equations

How the SAT tests this, with worked examples — plus the 2 skills it covers. Free to read, no account needed.

A circle is the set of points a fixed distance from a fixed centre, and its standard equation says exactly that

{(x-h)^2 + (y-k)^2 = r^2}

where

(h, k)

is the centre and

r

the radius.

Read the signs carefully. The equation contains

(x-h)

, so

(x-3)^2

puts the centre at

x = 3

, while

(x+3)^2

puts it at

x = -3

. The right-hand side is

r^2

, not

r

, so

= 9

means a radius of

3

When a question gives an expanded equation such as

x^2 + y^2 - 6x + 4y = 12

, it is asking you to complete the square in

x

and in

y

separately to recover the standard form.

Take half the coefficient of

x

, square it, and add it to both sides; then repeat for

y

. Everything else follows from reading

(h, k)

and

r

once the equation is in standard form.

Worked examples

Give the centre and radius of

(x-3)^2 + (y+1)^2 = 25

.
Matching the standard form,

h = 3

and

k = -1

, because

(y+1)

(y-(-1))

. Since

r^2 = 25

, the radius is

5

. So the centre is

(3, -1)

with radius

5

Find the centre of

x^2 + y^2 - 6x + 4y = 12

.
Complete the square in each variable

{(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4}

which gives

(x-3)^2 + (y+2)^2 = 25

, so the centre is

(3, -2)

and the radius

5

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