What is the Equation of the Circle that Passes Through Given Points and Cuts Through a Specific Line?

In geometry, determining the equation of a circle that passes through certain points and has its center on a given line is a common problem. Here's a detailed walkthrough of solving such a problem.

Problem Statement

The task is to find the equation of the circle that passes through the points 1,3 and 2,3 and whose center lies on the line 2x - 3y - 1 0.

Step-by-Step Solution

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

x2 - h2 y2 - k2 r2

Step 1: Use Given Points

Substitute the points (1, 3) and (2, 3) into the circle equation.

For point (1, 3): 1 - h2 3 - k2 r2 ... (1)

For point (2, 3): 2 - h2 3 - k2 r2 ... (2)

Step 2: Set Equations Equal

Since both equations equal r2, set them equal to each other:

1 - h2 3 - k2 2 - h2 3 - k2

Step 3: Simplify the Equation

The terms 3 - k2 cancel out:

1 - h2 2 - h2

Expanding both sides:

1 - 2h h2 4 - 4h h2

Cancelling h2 from both sides:

1 - 2h 4 - 4h

Rearranging the equation:

2h 3 implies h 3/2

Step 4: Find the y-coordinate (k)

The center (h, k) lies on the line 2x - 3y - 1 0. Substituting h 3/2 into the line equation:

2(3/2) - 3k - 1 0

3 - 3k - 1 0 implies 3k 2 implies k -2/3

Step 5: Find the Radius (r)

Now, the center of the circle is (3/2, -2/3). Using one of the points, say (1, 3), to find r:

(1 - 3/2)2 (3 2/3)2 r2

(-1/2)2 (11/3)2 r2

(1/4) (121/9) r2

36 * (1/4) 36 * (121/9) 493/36

r2 493/36

Step 6: Write the Equation of the Circle

With the center (3/2, -2/3), the equation is:

(x - 3/2)2 (y 2/3)2 493/36

Multiplying through by 36 to eliminate the fraction:

36(x - 3/2)2 36(y 2/3)2 493

Final Equation

The equation of the circle is:

36(x - 3/2)2 36(y 2/3)2 493