To solve the problem, we need to find the equation of a circle that passes through the points (-2, 5) and (0, 0), and intersects the given circle \(x^2 + y^2 - 4x + 3y - 2 = 0\) orthogonally.
### Step-by-Step Solution:
1. **Equation of the Circle:**
The general equation of a circle can be expressed as:
\[
x^2 + y^2 + 2gx + 2fy + c = 0
\]
Since the circle passes through the points (-2, 5) and (0, 0), we can substitute these points into the equation to form equations in terms of \(g\), \(f\), and \(c\).
2. **Substituting Point (0, 0):**
Substituting (0, 0) into the circle equation:
\[
0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \implies c = 0
\]
Thus, the equation simplifies to:
\[
x^2 + y^2 + 2gx + 2fy = 0
\]
3. **Substituting Point (-2, 5):**
Now substituting (-2, 5):
\[
(-2)^2 + (5)^2 + 2g(-2) + 2f(5) = 0
\]
This simplifies to:
\[
4 + 25 - 4g + 10f = 0 \implies -4g + 10f + 29 = 0 \quad \text{(Equation 1)}
\]
4. **Orthogonality Condition:**
For the circles to intersect orthogonally, the following condition must hold:
\[
2g_1g_2 + 2f_1f_2 = c_1c_2 + c_3
\]
Here, for the given circle \(x^2 + y^2 - 4x + 3y - 2 = 0\), we have \(g_2 = -2\), \(f_2 = \frac{3}{2}\), and \(c_2 = -2\).
Substituting into the orthogonality condition:
\[
2g(-2) + 2f\left(\frac{3}{2}\right) = (0)(-2) + 0
\]
This simplifies to:
\[
-4g + 3f = 0 \quad \text{(Equation 2)}
\]
5. **Solving the System of Equations:**
Now we have two equations:
- From Equation 1: \(-4g + 10f + 29 = 0\)
- From Equation 2: \(-4g + 3f = 0\)
We can express \(g\) in terms of \(f\) from Equation 2:
\[
-4g = -3f \implies g = \frac{3}{4}f
\]
Substituting \(g\) into Equation 1:
\[
-4\left(\frac{3}{4}f\right) + 10f + 29 = 0
\]
Simplifying gives:
\[
-3f + 10f + 29 = 0 \implies 7f + 29 = 0 \implies f = -\frac{29}{7}
\]
Now substituting \(f\) back to find \(g\):
\[
g = \frac{3}{4}\left(-\frac{29}{7}\right) = -\frac{87}{28}
\]
6. **Equation of the Circle:**
Now substituting \(g\) and \(f\) back into the circle equation:
\[
x^2 + y^2 - \frac{87}{14}x - \frac{58}{7}y = 0
\]
Multiplying through by 28 to eliminate fractions:
\[
28x^2 + 28y^2 - 174x - 232y = 0
\]
Rearranging gives:
\[
2x^2 + 2y^2 - 87x - 116y = 0
\]
### Final Equation:
The equation of the circle is:
\[
2x^2 + 2y^2 - 87x - 116y = 0
\]
