A circle is the path of every point lying the same distance (the radius) from one center.
With center (a, b) and radius r it becomes (x − a)² + (y − b)² = r², straight from the distance formula.
An expanded general form turns readable again once you complete the square to recover the center and radius.
Drag the center and radius here to see the equation and the circle change together.
Equation of a Circle — Center & Radius
A circle is the set of points equidistant from a center. Knowing the center (a, b) and radius r determines the equation.
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Circle: points at distance r from center (a,b)
💡 Deriving the Equation
①Distance from (x, y) on circle to center (a, b) = r
②√((x-a)² + (y-b)²) = r
③Square both sides → (x-a)² + (y-b)² = r²
Standard Form & General Form
Standard Form
(x - a)² + (y - b)² = r²
Center (a, b), radius r
General Form
x² + y² + Dx + Ey + F = 0
Expanded: D=-2a, E=-2b, F=a²+b²-r²
📐 General → Standard
①Group x: (x + D/2)²
②Group y: (y + E/2)²
③r² = (D/2)² + (E/2)² - F
④Need r² > 0 for the circle to exist
Circle vs Line Relations
The distance d from the center to the line, compared with radius r, determines the relation.
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Distance d from center to line vs radius r
Circle–Line Position
d < r: two points, d = r: tangent, d > r: no intersection
d = distance from center to the line
Tangent Lines
Tangent at a Point on the Circle
x1·x + y1·y = r²
For a point (x₁,y₁) on x²+y²=r²
Tangent with Slope m
y = mx ± r√(1 + m²)
Line of slope m tangent to the origin-centered circle
Work It Out
Example 1
Find the equation of the circle with center (2, −1) and radius 3.
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Substitute a = 2, b = −1, r = 3 into the standard form (x − a)² + (y − b)² = r². Since b is negative, it becomes (y + 1).
(x − 2)² + (y + 1)² = 3²
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Evaluate the right side.
(x − 2)² + (y + 1)² = 9
▸ (x − 2)² + (y + 1)² = 9
The center coordinates enter the equation with flipped signs.
Example 2
Find the center and radius of the circle x² + y² − 6x + 4y − 12 = 0.
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Complete the square in x and in y.
(x − 3)² − 9 + (y + 2)² − 4 − 12 = 0
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Move the constants to the right to reach standard form.