Practice Equation of a Circle in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The standard form equation (x−h)2+(y−k)2=r2 describes a circle with center (h,k) and radius r in the coordinate plane.

A circle is the set of all points at the same distance (the radius) from a center point. The equation just says 'the distance from (x,y) to the center (h,k) equals r,' using the distance formula squared.

Showing a random 20 of 50 problems.

Example 1

medium
A diameter of a circle has endpoints (2,3) and (8,11). Find the equation.

Example 2

easy
Write the equation of a circle with center (0,0) and radius 7.

Example 3

medium
Write the equation of a circle centered at (2,−1) tangent to the line y=3.

Example 4

easy
Give the radius of (x−4)2+(y+2)2=49.

Example 5

hard
Find the equation of the circle passing through (1,0), (5,0), and (3,4).

Example 6

medium
Determine whether x2+y2−2x+4y+10=0 describes a real circle.

Example 7

hard
Determine whether the circles (x−1)2+(y−3)2=16 and (x−7)2+(y−3)2=4 intersect, and find the number of intersection points.

Example 8

hard
Find the points where the circle x2+y2=25 intersects the line y=x+1.

Example 9

medium
Find the equation of the circle centered at (1,2) passing through (4,6).

Example 10

easy
Give the center of (x+5)2+(y−1)2=9.

Example 11

easy
Does the point (3,4) lie on x2+y2=25?

Example 12

challenge
A circle is tangent to the x-axis at (4,0) with radius 3. Find its equation(s).

Example 13

challenge
Find all values of b for which the line y=x+b is tangent to the circle x2+y2=8.

Example 14

easy
What is the radius of x2+y2=36?

Example 15

hard
Determine the relationship (intersecting, tangent, or disjoint) between the circles (x−1)2+(y−1)2=4 and (x−5)2+(y−4)2=9.

Example 16

medium
Find the equation of the circle with endpoints of a diameter at (1,2) and (7,10).

Example 17

hard
Find the equation of the circle tangent to both the x- and y-axes in the first quadrant with radius r=6.

Example 18

easy
Does (2,−1) lie on the circle (x−2)2+(y+1)2=0.0001?

Example 19

medium
Is x2+y2+2x+2y+2=0 a circle? If so, give center and radius.

Example 20

medium
Find the equation of the circle centered at the origin and passing through (5,12).
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