Equations of a Circle
SAT Math — Geometry & Coordinate Geometry
Standard Form
General Form
Completing the Square
Tangent Lines
1. Introduction — What Is a Circle?
Definition
A circle is the set of all points in a plane that are equidistant from a fixed point called the center.
| Term | Meaning | Notation |
|---|---|---|
| Center | The fixed point | $(h, k)$ |
| Radius | Constant distance from center to any point on the circle | $r$ |
| Diameter | Twice the radius; longest chord | $d = 2r$ |
| Standard Form | $(x-h)^2+(y-k)^2=r^2$ | Center-radius form |
| General Form | $x^2+y^2+Dx+Ey+F=0$ | Expanded form |
Why circle equations matter on the SAT: They connect geometry with algebra and appear in coordinate geometry problems. You must be able to identify center/radius, convert between forms, and solve intersection problems.
Interactive visualization — available in the full lesson
Drag point P around the circle — the distance to the center stays constant.
2. Standard Form (Center-Radius Form)
$$(x - h)^2 + (y - k)^2 = r^2$$
Center: $(h, k)$ | Radius: $r = \sqrt{r^2}$
Critical Sign Rule
- $(x - 3)^2$ means $h = +3$ (subtract positive → center coordinate is positive)
- $(x + 3)^2 = (x - (-3))^2$ means $h = -3$ (add → center coordinate is negative)
- Same rule applies to $k$ in the $y$-term.
Worked Examples — Reading Center & Radius
| # | Equation | Center $(h,k)$ | $r^2$ | Radius $r$ |
|---|---|---|---|---|
| 11 | $(x-3)^2+(y-5)^2=16$ | $(3,\,5)$ | 16 | $\sqrt{16}=4$ |
| 12 | $(x+2)^2+(y-4)^2=25$ | $(-2,\,4)$ | 25 | $\sqrt{25}=5$ |
| 13 | $(x-7)^2+(y+1)^2=49$ | $(7,\,-1)$ | 49 | $\sqrt{49}=7$ |
| 14 | $(x+5)^2+(y+3)^2=36$ | $(-5,\,-3)$ | 36 | $\sqrt{36}=6$ |
| 15 | $x^2+y^2=9$ | $(0,\,0)$ | 9 | $\sqrt{9}=3$ |
| 16 | $(x-1)^2+y^2=100$ | $(1,\,0)$ | 100 | $\sqrt{100}=10$ |
| 17 | $x^2+(y-6)^2=144$ | $(0,\,6)$ | 144 | $\sqrt{144}=12$ |
| 18 | $(x-2)^2+(y-3)^2=20$ | $(2,\,3)$ | 20 | $\sqrt{20}=2\sqrt{5}$ |
🚫 Common Mistake #150
Forgetting to take the square root. If the equation says $= 16$, the radius is $\sqrt{16} = 4$, not 16.