To find the equation of the circle that passes through the points where the lines \(2x - 5y + 1 = 0\) and \(10x - 4y - 3 = 0\) meet the coordinate axes, we will follow these steps:
### Step 1: Find the points where the first line intersects the axes.
1. **Finding the y-intercept** (set \(x = 0\)):
\[
2(0) - 5y + 1 = 0 \implies -5y + 1 = 0 \implies y = \frac{1}{5}
\]
So, the point is \(A(0, \frac{1}{5})\).
2. **Finding the x-intercept** (set \(y = 0\)):
\[
2x - 5(0) + 1 = 0 \implies 2x + 1 = 0 \implies x = -\frac{1}{2}
\]
So, the point is \(B(-\frac{1}{2}, 0)\).
### Step 2: Find the points where the second line intersects the axes.
1. **Finding the y-intercept** (set \(x = 0\)):
\[
10(0) - 4y - 3 = 0 \implies -4y - 3 = 0 \implies y = -\frac{3}{4}
\]
So, the point is \(C(0, -\frac{3}{4})\).
2. **Finding the x-intercept** (set \(y = 0\)):
\[
10x - 4(0) - 3 = 0 \implies 10x - 3 = 0 \implies x = \frac{3}{10}
\]
So, the point is \(D(\frac{3}{10}, 0)\).
### Step 3: List the points.
The points we have found are:
- \(A(0, \frac{1}{5})\)
- \(B(-\frac{1}{2}, 0)\)
- \(C(0, -\frac{3}{4})\)
- \(D(\frac{3}{10}, 0)\)
### Step 4: Set up the equation of the circle.
The general equation of a circle is:
\[
x^2 + y^2 + 2gx + 2fy + c = 0
\]
### Step 5: Substitute the points into the circle equation.
1. **Using point A(0, \frac{1}{5})**:
\[
0^2 + \left(\frac{1}{5}\right)^2 + 2g(0) + 2f\left(\frac{1}{5}\right) + c = 0
\]
\[
\frac{1}{25} + \frac{2f}{5} + c = 0 \quad \text{(Equation 1)}
\]
2. **Using point B(-\frac{1}{2}, 0)**:
\[
\left(-\frac{1}{2}\right)^2 + 0^2 + 2g\left(-\frac{1}{2}\right) + 2f(0) + c = 0
\]
\[
\frac{1}{4} - g + c = 0 \quad \text{(Equation 2)}
\]
3. **Using point C(0, -\frac{3}{4})**:
\[
0^2 + \left(-\frac{3}{4}\right)^2 + 2g(0) + 2f\left(-\frac{3}{4}\right) + c = 0
\]
\[
\frac{9}{16} - \frac{3f}{2} + c = 0 \quad \text{(Equation 3)}
\]
4. **Using point D(\frac{3}{10}, 0)**:
\[
\left(\frac{3}{10}\right)^2 + 0^2 + 2g\left(\frac{3}{10}\right) + 2f(0) + c = 0
\]
\[
\frac{9}{100} + \frac{3g}{5} + c = 0 \quad \text{(Equation 4)}
\]
### Step 6: Solve the equations.
We will solve these equations step by step to find \(g\), \(f\), and \(c\).
1. From Equation 1:
\[
c = -\frac{1}{25} - \frac{2f}{5}
\]
2. Substitute \(c\) from Equation 1 into Equation 2:
\[
\frac{1}{4} - g - \left(-\frac{1}{25} - \frac{2f}{5}\right) = 0
\]
Simplifying gives:
\[
\frac{1}{4} - g + \frac{1}{25} + \frac{2f}{5} = 0
\]
3. Substitute \(c\) from Equation 1 into Equation 3:
\[
\frac{9}{16} - \frac{3f}{2} - \left(-\frac{1}{25} - \frac{2f}{5}\right) = 0
\]
Simplifying gives:
\[
\frac{9}{16} + \frac{1}{25} - \frac{3f}{2} - \frac{2f}{5} = 0
\]
4. Continue solving these equations to find the values of \(g\), \(f\), and \(c\).
### Final Step: Write the equation of the circle.
Once \(g\), \(f\), and \(c\) are found, substitute them back into the general equation of the circle to get the final equation.
