How to Find the Equation of a Circle Given Two Points and the Radius

When dealing with geometric problems, particularly in analytic geometry, you may need to find the equation of a circle that passes through two given points and has a specified radius. This article covers the step-by-step process to solve such a problem using the distance formula and system of equations.

The Step-by-Step Process

Here's a detailed breakdown of the process:

Step 1: Identify the Points and Radius

Let the two given points on the circle be ( A(x_1, y_1) ) and ( B(x_2, y_2) ), and the given radius be ( r ).

Step 2: Midpoint and Distance

Calculate the midpoint ( M ) of the segment connecting points ( A ) and ( B ):

[ M left( frac{x_1 x_2}{2}, frac{y_1 y_2}{2} right) ]

Calculate the distance ( d ) between points ( A ) and ( B ):

[ d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2} ]

Step 3: Check the Radius Condition

For a circle to exist with radius ( r ) that passes through both points, the distance ( d ) must be less than or equal to ( 2r ).

Step 4: Set Up the Circle Equation

The general equation of a circle with center ( (h, k) ) and radius ( r ) is:

[ (x - h)^2 (y - k)^2 r^2 ]

Step 5: Find Possible Centers

The center ( (h, k) ) must be located on the perpendicular bisector of the line segment ( AB ). The midpoint ( M ) is one point on this bisector.

The distance from the center ( (h, k) ) to either point ( A ) or ( B ) should equal the radius ( r ).

Step 6: Use the Circle Equation

You can set up the equations based on the distance from the center to the points:

[ (x_1 - h)^2 (y_1 - k)^2 r^2 ]

[ (x_2 - h)^2 (y_2 - k)^2 r^2 ]

Step 7: Solve the System of Equations

To find the possible centers ( (h, k) ) of the circle, you need to solve this system of equations involving ( h ) and ( k ).

This system of equations can be solved using algebraic methods or numerical techniques, depending on the specific values of ( x_1, y_1, x_2, y_2 ), and ( r ).

Example

Suppose the points are ( A(1, 2) ) and ( B(4, 6) ), and the radius ( r 3 ).

Midpoint ( M left( frac{1 4}{2}, frac{2 6}{2} right) (2.5, 4) )

Distance ( d sqrt{(4 - 1)^2 (6 - 2)^2} sqrt{9 16} 5 )

Since ( d 5 ) is less than ( 2r 6 ), a circle can exist.

Set up the equations:

[ (1 - h)^2 (2 - k)^2 9 ]

[ (4 - h)^2 (6 - k)^2 9 ]

You can solve these equations to find the possible centers of the circle.

Independence of the Center and Radius on Given Points

When a circle has center ( (a, b) ) and radius ( r ), for any point ( (x, y) ) on the circle, the distance from ( (x, y) ) to ( (a, b) ) is ( r ). The distance formula is:

[ sqrt{(x - a)^2 (y - b)^2} r ]

Squaring both sides:

[ (x - a)^2 (y - b)^2 r^2 ]

This is the equation of the circle.

Conclusion

Understanding and applying the process described above will enable you to confidently find the equation of a circle that passes through two given points and has a specified radius. This knowledge is not only useful in geometric problems but also in various fields such as computer graphics, engineering, and physics where circles play a significant role.