Write and interpret the standard equation of a circle in the coordinate plane.
A circle is the set of all points equidistant from a fixed center point. Using the distance formula, we derive the standard equation of a circle with center (h, k) and radius r.
(x−h)2+(y−k)2=r2
Converting General Form to Standard Form
1Group x-terms and y-terms together.
2Complete the square for both x and y.
3Add the same values to both sides of the equation.
4Write in standard form and identify center (h, k) and radius r.
x2+y2−6x+4y−12=0
(x2−6x+9)+(y2+4y+4)=12+9+4=25
(x−3)2+(y+2)2=25⟹center (3,−2),r=5
💡
Remember This!
Completing the square: take half the coefficient of the linear term, square it, and add it to both sides. For x² − 6x: take (−6/2)² = 9 and add 9 to both sides.
✏️ Try It!
What is the center and radius of the circle (x + 1)² + (y − 4)² = 36?