Chapter 4 · Geometry and Equations · Concept 3
4-16 Equation of a Circle (1)
A circle is nothing more than every point sitting the same distance from one middle point. Write that down with the distance formula and you get the equation of a circle — and it comes in two outfits, one that shows you the centre and one that hides it.
Guiding question
If you know where the middle is and how far it reaches, can you write the circle down?
Learning outcome. Write the equation of a circle in standard and general form, and change one into the other.
The same circle, said two ways
STANDARD FORM — centre (a, b), radius r
(x − a)2 + (y − b)2 = r2
and if the middle is the origin, a and b are both 0, so it shrinks to x² + y² = r².
GENERAL FORM — brackets multiplied out
x2 + y2 + lx + my + n = 0
Same circle. But the centre and the radius are now buried, and completing the square is how you dig them out.
The minus sign is part of the form.
The form is (x − a)², so whatever is inside gets its sign flipped to give the centre. (x − 3)² → centre at +3. (x + 3)² is really (x − (−3))², so → centre at −3. And = 25 on the right is r2, so the radius is 5, not 25.
Where the equation comes from.
A point (x, y) is on the circle exactly when its distance from the centre (a, b) is r. The distance formula says √((x − a)² + (y − b)²) = r. Square both sides to lose the root and you have the standard form. That is the entire derivation.
Do the x letters first, then the y letters, then sweep the loose numbers to the other side. Watch what each step leaves behind — that leftover is where most marks go missing.
x² + y² − 6x + 2y + 6 = 0
the x parts
x² − 6x
the y parts
y² + 2y
put it all back together
…
tidy the numbers to the other side
…
The general form hides both — nothing here says where the middle is.
2. Show that x² + y² − 6x + 2y + 6 = 0 is a circle, and find its centre and radius.
- 1
Gather the x letters and the y letters into two separate piles.
(x2 − 6x) + (y2 + 2y) + 6 = 0The plain number 6 belongs to neither pile. Leave it alone for now.
Work with a partner. One of you completes the square; the other checks it by going backwards — multiply the brackets out again and see whether you land back on the question. If the middle numbers do not match, someone forgot a leftover.
1. Write down the circle.
2. Show each is a circle, and find the centre and radius.
(b) and (d) are missing a piece on purpose.
In (b) there is no y term at all, so y² is already a finished square — nothing to complete, and the centre sits on the x-axis. In (d) there is no plain number, so the right-hand side is built only from the two leftovers. Neither is a special rule; both fall straight out of the same method.
A surveillance drone flies a circular patrol. Two fixed checkpoints on the route are recorded at A(2, 5) and B(8, 1), and the control tower — the centre of the circle — is known to sit somewhere on the road x + y = 8. Find the equation of the patrol circle.
Writing the radius where r² belongs.
Radius 5 gives = 25 on the right. Going the other way, = 25 means the radius is 5. The right-hand side is always the SQUARE.
Reading the centre without flipping the sign.
(x + 3)² means the centre is at −3. The form is (x − a)², so whatever sits inside gets reversed.
Forgetting the leftover when completing the square.
x² − 6x is (x − 3)² − 9, not (x − 3)². Miss the −9 and you get the right centre with the wrong radius, and the working looks perfectly tidy.
Halving the wrong number.
You halve the coefficient of x, not the constant. In x² − 6x + 6 you halve the −6, and the +6 waits its turn on the other side.
Not checking that the right-hand side is positive.
"Show that it is a circle" is asking exactly this. If the number came out negative or zero there would be no circle at all — which is the whole of lesson 4-19.
Sign In to Keep Going
You've reached the end of the free preview. Sign in — and upgrade to Plus or Pro — to keep going with Mathematics.
Full Access
Unlock all 105 lessons