There are 2 circles passing through points A(-1, 2) and B(2, 3) having radius `sqrt(5)`. Then the length of intercept on axis of the circle intersecting x-axis is :

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to find the length of the intercept on the x-axis of the circle that passes through the points A(-1, 2) and B(2, 3) with a radius of \(\sqrt{5}\).

### Step-by-Step Solution:

1. **Identify the center of the circle**:
Let the center of the circle be \(C(a, b)\). The distances from the center to points A and B must equal the radius \(\sqrt{5}\).

2. **Set up the equations based on the distances**:
The distance \(CA\) can be expressed as:
\[ CA^2 = (a + 1)^2 + (b - 2)^2 = 5 \]
The distance \(CB\) can be expressed as:
\[ CB^2 = (a - 2)^2 + (b - 3)^2 = 5 \]

3. **Expand both equations**:
Expanding the first equation:
\[ (a + 1)^2 + (b - 2)^2 = 5 \implies a^2 + 2a + 1 + b^2 - 4b + 4 = 5 \implies a^2 + b^2 + 2a - 4b = 0 \quad \text{(Equation 1)} \]
Expanding the second equation:
\[ (a - 2)^2 + (b - 3)^2 = 5 \implies a^2 - 4a + 4 + b^2 - 6b + 9 = 5 \implies a^2 + b^2 - 4a - 6b + 8 = 0 \quad \text{(Equation 2)} \]

4. **Subtract Equation 1 from Equation 2**:
\[ (a^2 + b^2 - 4a - 6b + 8) - (a^2 + b^2 + 2a - 4b) = 0 \]
This simplifies to:
\[ -6a - 2b + 8 + 4 = 0 \implies -6a - 2b + 12 = 0 \implies 3a + b = 6 \quad \text{(Equation 3)} \]

5. **Substitute Equation 3 into Equation 1**:
From Equation 3, we can express \(b\) in terms of \(a\):
\[ b = 6 - 3a \]
Substituting into Equation 1:
\[ a^2 + (6 - 3a)^2 + 2a - 4(6 - 3a) = 0 \]
Expanding this gives:
\[ a^2 + (36 - 36a + 9a^2) + 2a - 24 + 12a = 0 \]
Simplifying:
\[ 10a^2 - 22a + 12 = 0 \]

6. **Solve the quadratic equation**:
Using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{22 \pm \sqrt{(-22)^2 - 4 \cdot 10 \cdot 12}}{2 \cdot 10} \]
\[ = \frac{22 \pm \sqrt{484 - 480}}{20} = \frac{22 \pm 2}{20} \]
This gives:
\[ a = \frac{24}{20} = 1.2 \quad \text{or} \quad a = \frac{20}{20} = 1 \]

7. **Find corresponding values of \(b\)**:
For \(a = 1.2\):
\[ b = 6 - 3(1.2) = 2.4 \]
For \(a = 1\):
\[ b = 6 - 3(1) = 3 \]

8. **Determine the equations of the circles**:
The centers are \(C(1.2, 2.4)\) and \(C(1, 3)\). The equations of the circles are:
\[ (x - 1.2)^2 + (y - 2.4)^2 = 5 \quad \text{and} \quad (x - 1)^2 + (y - 3)^2 = 5 \]

9. **Find the x-intercepts**:
Set \(y = 0\) in both equations and solve for \(x\):
For the first circle:
\[ (x - 1.2)^2 + (0 - 2.4)^2 = 5 \implies (x - 1.2)^2 + 5.76 = 5 \implies (x - 1.2)^2 = -0.76 \quad \text{(no real solution)} \]
For the second circle:
\[ (x - 1)^2 + (0 - 3)^2 = 5 \implies (x - 1)^2 + 9 = 5 \implies (x - 1)^2 = -4 \quad \text{(no real solution)} \]

10. **Conclusion**:
Since both circles do not intersect the x-axis, the length of the intercept on the x-axis is \(0\).

### Final Answer:
The length of the intercept on the x-axis of the circle intersecting the x-axis is \(0\).