Conic Sections: Parabolas & Circles
Write and graph parabolas and circles: read center-radius form, locate vertex, focus, and directrix, and complete the square to reveal hidden centers.
1 Circles: center-radius form
A circle is the set of points at distance r from a center (h, k). The equation says exactly that, via the distance formula.
Center (h, k), radius r. Watch the signs: (x − 2)2 means h = +2; (y + 3)2 means k = −3. The signs flip when you read the center.
2 Parabolas: vertex, focus, directrix
A parabola is the set of points equidistant from a focus and a directrix line. The vertex sits halfway between them.
Horizontal: x = a(y − k)2 + h · focus (h + 14a, k) · directrix x = h − 14a
The squared variable sets the axis: x2 → vertical axis (opens up if a > 0, down if a < 0); y2 → horizontal axis (opens right if a > 0, left if a < 0). The focal length p = 1/(4a).
3 Completing the square
General form hides the center and vertex. Completing the square reveals them.
Group: (x2 + 4x) + (y2 − 6y) = 12. Complete: (x2 + 4x + 4) + (y2 − 6y + 9) = 12 + 4 + 9.
So (x + 2)2 + (y − 3)2 = 25: center (−2, 3), radius 5.
Complete the square in x: y = (x2 − 4x + 4) + 5 − 4 = (x − 2)2 + 1.
Vertex (2, 1); a = 1 > 0 so it opens up.
The shortcut is completing the square in disguise — use it to check your work, not to skip learning the method.
4 Worked examples
Read, convert, and locate — each claim verified independently.
Find the center and radius of (x − 2)2 + (y + 3)2 = 25.
Center (2, −3) (signs flip), radius 5 (√25).
Find the center and radius of x2 + y2 + 4x − 6y − 12 = 0.
(x + 2)2 + (y − 3)2 = 25: center (−2, 3), radius 5.
Find the vertex of y = x2 − 4x + 5.
y = (x − 2)2 + 1, so the vertex is (2, 1).
Vertex (0, 0), a = 1, so p = 1/(4·1) = 0.25. Focus: (0, 0.25).
Directrix: y = k − p = 0 − 0.25, so y = −0.25.
Which way does x = −2(y − 1)2 + 3 open?
y is squared → horizontal axis; a = −2 < 0 → opens left.
Find the vertex of y = 2x2 + 8x + 5.
y = 2(x2 + 4x) + 5 = 2(x + 2)2 − 8 + 5 = 2(x + 2)2 − 3: vertex (−2, −3).
5 Common mistakes
Two sign errors that haunt conic problems.
Wrong: copying the signs straight across.
Right: the form is (x − h)2 + (y − k)2, so (x − 2)2 gives h = +2 and (y + 3)2 gives k = −3: center (2, −3).
Wrong: saying y = −x2 + 4 opens left or right.
Right: x2 → the axis is vertical: up (a > 0) or down (a < 0). Here a = −1, so it opens down. y2 is what opens left/right.
6 Key vocabulary
Words to know
- Center-radius form — (x − h)2 + (y − k)2 = r2.
- Vertex — the turning point of a parabola, halfway between focus and directrix.
- Focus — the fixed point a parabola curves around; distance p = 1/(4a) from the vertex.
- Directrix — the fixed line; every parabola point is equidistant from focus and directrix.
- Completing the square — rewriting ax2 + bx as a(x + b/2a)2 − (constant).
7 Quick check
Try these before moving on — click to reveal each answer.
8 Next steps
Now drill the skill with endless randomized problems.