What is the equation of a circle that is concentric with x2 y2 - 8x 12y 15 0 and passes through (5, 4)?

To find the equation of a circle that is concentric with the given circle and passes through the point (5, 4), we need to first determine the center and radius of the original circle. Let's go through this step-by-step.

Step 1: Rewrite the Original Circle's Equation

The given equation of the circle is:

x2   y2 - 8x   12y   15  0

We can rearrange this to the standard form by completing the square.

Completing the Square for x:

Taking the x terms: x2 - 8x.

To complete the square, take half of -8 and square it: (-4)^2 16.

Rewrite: x2 - 8x (x - 4)^2 - 16

Completing the Square for y:

Taking the y terms: y2 12y.

To complete the square, take half of 12 and square it: 6^2 36.

Rewrite: y2 12y (y 6)^2 - 36

Step 2: Substitute Back into the Equation

Substituting these completed squares back into the equation gives:

(x - 4)^2 - 16   (y   6)^2 - 36   15  0

Simplifying this:

(x - 4)^2   (y   6)^2 - 37  0

This simplifies to:

(x - 4)^2   (y   6)^2  37

Step 3: Identify the Center and Radius

From the equation (x - 4)2 (y 6)2 37, we can see:

The center of the circle is (4, -6). The radius r is sqrt{37}.

Step 4: Find the Equation of the New Circle

Since the new circle is concentric with the original circle, it will have the same center (4, -6). We need to find the radius of this new circle which must pass through the point (5, 4).

Step 4.1: Calculate the Radius

The distance d from the center (4, -6) to the point (5, 4) can be calculated using the distance formula:

d  sqrt{(5 - 4)^2   (4 - (-6))^2}  sqrt{1^2   10^2}  sqrt{1   100}  sqrt{101}

Step 5: Write the Equation of the New Circle

The equation of the new circle with center (4, -6) and radius sqrt{101} is:

(x - 4)^2   (y   6)^2  101

Final Answer: The equation of the circle that is concentric with the original circle and passes through the point (5, 4) is (x - 4)2 (y 6)2 101.