SME
Grade 10 Math · Problem type
Center and radius from the general form
Use this for “Find the center and radius of x² + y² − 6x + 4y − 12 = 0.”
Diagram
(x − 3)² + (y + 2)² = 25
Center (3, −2)
radius 5.
This is an example. Type your own values to start a new problem.
x²
+
y²
+
(
D
)
x
+
E
y
+
(
F
)
=
0
(x
−
h
)²
+
(y
−
(
k
)
)²
=
r
2
Completing the square in x and in y turns the first line into the second.
Formulas
h
=
−
D
2
3
=
−
−6
2
k
=
−
E
2
−2
=
−
4
2
r
=
√
h
2
+
k
2
−
F
5
=
√(
3
2
+
(−2)
2
−
(−12))
Assumptions
•
Complete the square in x and in y: x² + Dx = (x + D/2)² − (D/2)².
•
So the center is (−D ÷ 2, −E ÷ 2) and r² = h² + k² − F.
•
When h² + k² − F is 0 or less, the equation has no circle.
Step-by-step
WE KNOW
D = −6, E = 4, F = −12
FIND
center, x (h), center, y (k), radius (r)
Step 1 · Find center, x (h)
Use h = −D ÷ 2
Half the x coefficient, with its sign changed, completes the square.
h
=
−
−6
2
h
=
3
Step 2 · Find center, y (k)
Use k = −E ÷ 2
Half the y coefficient, with its sign changed, completes the square.
k
=
−
4
2
k
=
−2
Step 3 · Find radius (r)
Use r = √(h² + k² − F)
Move
the
squares’
extra
terms
and
F
to
the
right
side:
that
is
r
2
.
r
=
√(
3
2
+
(−2)
2
−
(−12))
r
=
√(9
+
4
−
(−12))
r
=
√
25
r
=
5
Check
Put the numbers back in. Both sides should match.
3
=
−
−6
2
✓
−2
=
−
4
2
✓
5
=
√(
3
2
+
(−2)
2
−
(−12))
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Pythagorean theorem and distance between points
Solving quadratic equations: factoring, square roots, completing the square and the quadratic formula