Practice Equation of a Circle in Math

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

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Quick Recap

The standard form equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 describes a circle with center (h,k)(h, k) and radius rr 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)(x, y) to the center (h,k)(h, k) equals rr,' using the distance formula squared.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

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

Example 4

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

Example 5

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

Example 6

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

Example 7

hard
Determine whether the circles (x−1)2+(y−3)2=16(x-1)^2 + (y-3)^2 = 16 and (x−7)2+(y−3)2=4(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=25x^2 + y^2 = 25 intersects the line y=x+1y = x + 1.

Example 9

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

Example 10

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

Example 11

easy
Does the point (3,4)(3,4) lie on x2+y2=25x^2+y^2=25?

Example 12

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

Example 13

challenge
Find all values of bb for which the line y=x+by = x + b is tangent to the circle x2+y2=8x^2 + y^2 = 8.

Example 14

easy
What is the radius of x2+y2=36x^2 + y^2 = 36?

Example 15

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

Example 16

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

Example 17

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

Example 18

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

Example 19

medium
Is x2+y2+2x+2y+2=0x^2 + y^2 + 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)(5, 12).
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