Equations of Circles

Geometry · Axiom Academy

Writing and interpreting circle equations in the coordinate plane — the center, the radius, and how to read them off. 1. A Circle Is Every Point at Distance r A circle is the set of all points (x, y) that sit a fixed distance r from a center (h, k) . Apply the distance formula and square both sides, and that single condition becomes the standard form of a circle. Distance from the center to any point on the circle is r Square both sides → standard form. Center (h, k) , radius r . 2. Reading the Center and Radius The center hides inside the subtraction signs, and the radius hides under a square . Match an equation against (x-h)^2 + (y-k)^2 = r^2 and read both off — but watch the signs and remember to take the square root. Read h and k as the numbers that make each bracket a true subtraction. (x + 4) is (x - (-4)) , so h = -4 . A plus inside means a negative coordinate. (x + 4)^2 + (y - 1)^2 gives center (-4, 1) , not (4, 1) . The right side is r^2 , not r . If r^2 = 49 then r = 7 — always take the square root. When the center is (0, 0) the brackets vanish: x^2 + y^2 = r^2 . Write the equation of a circle with center (3, -2) and radius 5 . Substitute h = 3 , k = -2 , r = 5 into the standard form: (x - 3)^2 + (y - (-2))^2 = 5^2 . Example: identify center and radius Find the center and radius of (x + 4)^2 + (y - 1)^2 = 49 . Rewrite (x+4) as (x-(-4)) , so h = -4 ; the (y-1) bracket gives k = 1 ; and r^2 = 49 gives r = 7 . 3. Two Forms, and Completing the Square

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