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Determine the equation of a circle from its centre and radius

The equation of a circle with centre (h,k)(h,k) and radius r>0r>0 is (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2, expressing that every point (x,y)(x,y) on the circle is a fixed distance rr from the centre. This understanding connects the distance formula, coordinate geometry, and graphical features, including the importance of the signs in (x−h)(x-h) and (y−k)(y-k); it focuses on standard form rather than more advanced conic representations.

Detailed Explanation: Determine the equation of a circle from its centre and radius

A circle with centre (h,k)(h,k) and radius rr has equation

(x−h)2+(y−k)2=r2.(x-h)^2+(y-k)^2=r^2.

The signs are important: you subtract the coordinates of the centre. If a coordinate is negative, subtracting it produces a plus sign.

Example: Determine the equation of the circle with centre (−2,3)(-2,3) and radius 55.

  1. Identify the values:

h=−2,k=3,r=5h=-2,\qquad k=3,\qquad r=5

  1. Substitute them into the standard form:

(x−(−2))2+(y−3)2=52 (x-(-2))^2+(y-3)^2=5^2

  1. Simplify:

(x+2)2+(y−3)2=25 \boxed{(x+2)^2+(y-3)^2=25}

Therefore, the equation of the circle is

(x+2)2+(y−3)2=25.\boxed{(x+2)^2+(y-3)^2=25}.

Notice that the centre is (−2,3)(-2,3), even though the equation contains (x+2)(x+2). The plus sign appears because x−(−2)=x+2x-(-2)=x+2.

Learn by doing: Determine the equation of a circle from its centre and radius

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    Graphing Circles - Center Coordinate and Radius to Equation

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