Finding the Equations of a Circle with Given Conditions: A Step-by-Step Guide

In this article, we will explore how to find the equations of a circle given specific conditions. We will use the example of a circle with a radius of 10 units, which passes through a particular point and has the x-axis as a tangent. We will break down the problem into several steps and show you how to apply the concepts of circle equations and tangents to arrive at the solution.

Problem Statement

A circle has a radius of 10 units and passes through the point (5, -16). The x-axis is a tangent to this circle. Determine the possible equations of the circle under these conditions.

Step 1: Determine the Center of the Circle

Since the x-axis is a tangent to the circle, the distance from the center of the circle to the x-axis must equal the radius. If the center of the circle is at (h, k), then:

(k 10)

This gives us two possible values for k: 10 and -10.

Step 2: Use the Point that the Circle Passes Through

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

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

Substituting (r 10), we have:

( (x - h)^2 (y - k)^2 100 )

Case 1: k 10

Substituting (k 10):

( (5 - h)^2 (-16 - 10)^2 100 )

Calculating the second term:

( (5 - h)^2 (-26)^2 100 )

This simplifies to:

( (5 - h)^2 676 100 )

Rearranging gives:

( (5 - h)^2 100 - 676 )

( (5 - h)^2 -576 )

Since a square cannot equal a negative number, there are no valid circles for this case.

Case 2: k -10

Now substituting (k -10):

( (5 - h)^2 (-16 - (-10))^2 100 )

Calculating the second term:

( (5 - h)^2 (-6)^2 100 )

This simplifies to:

( (5 - h)^2 36 100 )

Rearranging gives:

( (5 - h)^2 100 - 36 )

( (5 - h)^2 64 )

Taking the square root:

( 5 - h 8 ) or ( 5 - h -8 )

This gives us two possible values for h:

( h -3 ) or ( h 13 )

Step 3: Write the Equations of the Circles

Now we can formulate the equations of the circles based on the centers we found:

For the center ((-3, -10)):

( (x 3)^2 (y 10)^2 100 )

For the center ((13, -10)):

( (x - 13)^2 (y 10)^2 100 )

Final Answer

The possible equations of the circles are:

( (x 3)^2 (y 10)^2 100 )

( (x - 13)^2 (y 10)^2 100 )