Equations of Circles in the Coordinate Plane

SAT Math · Axiom Academy

Equations of Circles in the Coordinate Plane Read off the center and radius, convert from general form by completing the square, and build a circle's equation from the information the SAT gives you. The standard form of a circle and how the center and radius are hidden inside it Reading the center (h, k) and radius r straight from an equation — and the sign trap that catches most students Converting from general form to standard form by completing the square (the #1 SAT circle skill) Building a circle's equation from a center and radius, a center and a point, or the endpoints of a diameter A circle is the set of all points a fixed distance r (the radius ) from a fixed point (h, k) (the center ). Applying the distance formula to that definition gives the standard form. Write the equation of the circle with center (3, -2) and radius 5 . : (x - 3)^2 + (y - (-2))^2 = 5^2 (x - 3)^2 + (y + 2)^2 = 25 Reading Off the Center and Radius If an equation is already in standard form, the center and radius are sitting right there — as long as you respect the sign trap. Identify the center and radius of (x + 1)^2 + (y - 4)^2 = 16 . (x - (-1))^2 + (y - 4)^2 = 4^2 General Form & Completing the Square The SAT often hides a circle inside its general form , where the squares are expanded: To find the center and radius, convert back to standard form by completing the square on the x -terms and the y -terms separately. Find the center and radius of x^2 + y^2 - 6x + 4y - 12 = 0 .

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