Solving Circle Equations: Finding a New Circle Passing Through Two Points and Concentric with Given Circles

In this article, we will explore how to find the equation of a new circle that passes through the center of one circle and is concentric with another. We'll walk through the steps and provide detailed calculations to help you understand the process.

Introduction to Circle Equations

Circles are fundamental geometric shapes in mathematics, and their equations help us understand and manipulate these shapes. The standard form of a circle's equation is:

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

where ((h, k)) is the center of the circle and (r) is the radius.

Step-by-Step Solution

Step 1: Finding the Center of the First Circle

Consider the equation of the first circle:

[ x^2 y^2 8x - 10y - 7 0 ]Completing the square for (x) and (y), we get:

[ x^2 8x y^2 - 10y 7 ] [ (x 4)^2 - 16 (y - 5)^2 - 25 7 ]Thus, the center of the first circle is ((-4, -5)).

Step 2: Finding the Center of the Second Circle

Now, consider the equation of the second circle:

[ 2x^2 2y^2 - 8x - 12y - 9 0 ]Divide the entire equation by 2:

[ x^2 y^2 - 4x - 6y - frac{9}{2} 0 ]Completing the square for (x) and (y), we get:

[ x^2 - 4x y^2 - 6y frac{9}{2} ] [ (x - 2)^2 - 4 (y - 3)^2 - 9 frac{9}{2} ]Thus, the center of the second circle is ((2, 3)).

Step 3: Writing the Equation of the New Circle

A circle that is concentric with the second circle will have the same center ((2, 3)). Our new circle must pass through the point ((-4, -5)).

The radius (r) of the new circle is the distance from ((2, 3)) to ((-4, -5)):

[ r sqrt{(2 - (-4))^2 (3 - (-5))^2} ] [ r sqrt{6^2 8^2} ] [ r sqrt{36 64} ] [ r sqrt{100} 10 ]Now, we can write the equation of the new circle:

[ (x - 2)^2 (y - 3)^2 100 ]The equation of the new circle is:

[ x^2 y^2 - 4x - 6y - 87 0 ]Conclusion

In this article, we demonstrated the process of finding the equation of a new circle that passes through a given point and is concentric with another circle. By completing the square, we determined the centers of both circles and then used the distance formula to find the radius of the new circle.

Related Keywords

- Circle equations - Concentric circles - Circle center finding