Section circles

Circles

Circles are your easiest win on Exam 4 — the equation is simpler than every other conic. More importantly, completing the square here is the exact same technique you'll need for ellipses and hyperbolas, so this section is your practice run.

⚡ Quick Summary

A circle is simply the set of all points that are the same distance from a center point — like tying a string to a pin, pulling it taut, and drawing a curve all the way around. That fixed distance is the radius, and the pin is the center.

Standard form of a circle

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

center $(h, k)$ , radius $r$

General form to standard form

x^2 + y^2 + Dx + Ey + F = 0 \;\Longrightarrow\; \left(x + \tfrac{D}{2}\right)^2 + \left(y + \tfrac{E}{2}\right)^2 = \tfrac{D^2 + E^2 - 4F}{4}

center $\left(-\tfrac{D}{2},\, -\tfrac{E}{2}\right)$ , radius $\tfrac{\sqrt{D^2 + E^2 - 4F}}{2}$

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The equation of a circle in standard form is $(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center and $r$ is the radius. The tricky part: the signs in the equation are *opposite* to the center's coordinates (so $(x + 3)$ means the center's $x$ -coordinate is $-3$ ), and the right side is $r^2$ , not $r$ itself — you need to take the square root to get the actual radius.

Sometimes circles are given in a messy expanded form like $x^2 + y^2 + 6x - 4y + 5 = 0$ . To make sense of it, you use completing the square — a rearranging trick that groups the $x$ -terms and $y$ -terms, adds the right numbers to create perfect squares, and transforms the mess back into standard form so the center and radius pop right out.

If a problem gives you the endpoints of a diameter, the center is just the midpoint (average the coordinates) and the radius is half the diameter's length. If you're given the center and a point on the circle, use the distance formula to find the radius, then plug everything into the standard-form equation.

Interactive Diagram

Key Formulas

Standard form of a circle

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

center $(h, k)$ , radius $r$

General form to standard form

x^2 + y^2 + Dx + Ey + F = 0 \;\Longrightarrow\; \left(x + \tfrac{D}{2}\right)^2 + \left(y + \tfrac{E}{2}\right)^2 = \tfrac{D^2 + E^2 - 4F}{4}

center $\left(-\tfrac{D}{2},\, -\tfrac{E}{2}\right)$ , radius $\tfrac{\sqrt{D^2 + E^2 - 4F}}{2}$

Distance formula (for finding radius)

r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}

where $(x_1, y_1)$ is any point on the circle

Key Takeaways

✓Standard form: $(x - h)^2 + (y - k)^2 = r^2$ — the center is $(h, k)$ and the right side is $r^2$ , not $r$
✓Signs flip: $(x + 3)$ means $h = -3$ — always take the opposite sign of what's inside the parentheses
✓To convert general form to standard form, complete the square for both the $x$ and $y$ groups
✓Given diameter endpoints, use the midpoint formula for the center and the distance formula for the radius

⚠️ Common Mistakes

✗Using the diameter as the radius — if the problem gives diameter $= 10$ , the radius is $5$ and $r^2 = 25$ , not $100$
✗Reading the right side of $(x-h)^2 + (y-k)^2 = r^2$ as the radius — the right side is $r^2$ ; take the square root to get the actual radius
✗Getting the center signs backwards — $(x + 3)$ means $h = -3$ , not $h = 3$ ; the signs in the equation are always opposite the center coordinates
✗Adding the completing-the-square number to the left side but not the right — if you add $9$ to one side, you MUST add $9$ to the other side too

Next: Section 10.1

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10.1 The Ellipse