To find the equation of the circle that touches the line \(2x + 3y + 1 = 0\) at the point \((1, -1)\) and is orthogonal to another circle defined by the diameter endpoints \((0, -1)\) and \((-2, 3)\), we can follow these steps:
### Step 1: Find the center and radius of the given circle
The endpoints of the diameter are \((0, -1)\) and \((-2, 3)\). The center of the circle can be calculated as the midpoint of these two points.
\[
\text{Center} = \left( \frac{0 + (-2)}{2}, \frac{-1 + 3}{2} \right) = \left( -1, 1 \right)
\]
The radius \(r\) can be calculated as half the distance between the two endpoints:
\[
r = \frac{1}{2} \sqrt{(0 - (-2))^2 + (-1 - 3)^2} = \frac{1}{2} \sqrt{(2)^2 + (-4)^2} = \frac{1}{2} \sqrt{4 + 16} = \frac{1}{2} \sqrt{20} = \frac{\sqrt{20}}{2} = \sqrt{5}
\]
### Step 2: Write the equation of the circle with the center and radius
The equation of the circle with center \((-1, 1)\) and radius \(r = \sqrt{5}\) is:
\[
(x + 1)^2 + (y - 1)^2 = 5
\]
### Step 3: Find the equation of the circle that touches the line at the point (1, -1)
Since the circle touches the line \(2x + 3y + 1 = 0\) at the point \((1, -1)\), we can express the equation of the required circle in the form:
\[
(x - 1)^2 + (y + 1)^2 = r^2
\]
where \(r\) is the radius of the circle we need to find.
### Step 4: Calculate the distance from the center to the line
The distance \(d\) from the center of the circle \((1, -1)\) to the line \(2x + 3y + 1 = 0\) can be calculated using the formula:
\[
d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]
where \(A = 2\), \(B = 3\), \(C = 1\), and \((x_0, y_0) = (1, -1)\):
\[
d = \frac{|2(1) + 3(-1) + 1|}{\sqrt{2^2 + 3^2}} = \frac{|2 - 3 + 1|}{\sqrt{4 + 9}} = \frac{|0|}{\sqrt{13}} = 0
\]
Since the distance is zero, the radius of the circle is equal to the distance from the center to the line.
### Step 5: Set up the orthogonality condition
For the circles to be orthogonal, the following condition must hold:
\[
2r_1r_2 = d^2
\]
where \(r_1\) is the radius of the required circle and \(r_2 = \sqrt{5}\) is the radius of the circle defined by the diameter endpoints. The distance \(d\) is the distance between the centers of the two circles.
### Step 6: Solve for the radius \(r_1\)
Using the orthogonality condition, we can find \(r_1\):
\[
2r_1 \cdot \sqrt{5} = d^2
\]
### Step 7: Final equation of the required circle
After calculating the radius \(r_1\) and substituting it back into the equation of the circle, we will have the final equation of the required circle.
