Find the equation of a circle given a point on the circle and a line containing the center.

Published on November 28, 2024

Geometry

The problem involves finding the equation of a circle given a point on the circle, the radius, and the fact that the center of the circle lies on a given line. Two possible circles satisfy the conditions, each with a different center, and the equations for both circles are derived.

Question

please solve the task.

Answer

Here’s how to solve this problem:

1. Understand the problem

We are given a point A(-1, 6) on a circle with radius

. The center of the circle (let’s call it M(a, b)) lies on the line

. We need to find the equation of the circle.

2. Set up the equations

The general equation of a circle is

. Since the radius is 5, our equation becomes:

Since the center M(a, b) lies on the line

, we have:

And since point A(-1, 6) lies on the circle, we can substitute its coordinates into the circle equation:

3. Solve for a and b

Substitute

into the equation derived from point A:

Simplify:

This is a quadratic equation. We can solve it by factoring:

So,

or

.

Case 1: a = 4

Center M(4, 6)
Case 2: a = -1

Center M(-1, 11)

4. Write the circle equations

Case 1: Center (4, 6)
Case 2: Center (-1, 11)

Therefore, there are two possible circle equations that satisfy the given conditions.

Final Answer: The final answer is

and

🚀 Try Free Math Help Now!🏠 Learn More Powerful Features