The intercepts made by the circle `x^(2)+y^(2)-5x-13y-14=0` on the x-axis and y-axis are respectively

Options

A. 9, 13
B. 5, 13
C. 9, 15
D. none

Answer

To find the intercepts made by the circle given by the equation \(x^2 + y^2 - 5x - 13y - 14 = 0\) on the x-axis and y-axis, we can follow these steps:

### Step 1: Rewrite the Circle Equation
The given equation of the circle is:
\[ x^2 + y^2 - 5x - 13y - 14 = 0 \]
We can compare this with the general form of a circle:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From this, we identify:
- \(2g = -5\) → \(g = -\frac{5}{2}\)
- \(2f = -13\) → \(f = -\frac{13}{2}\)
- \(c = -14\)

### Step 2: Find the X-axis Intercept
The formula for the x-axis intercept is:
\[ \text{Intercept on x-axis} = 2\sqrt{g^2 - c} \]
Substituting the values of \(g\) and \(c\):
\[ \text{Intercept on x-axis} = 2\sqrt{\left(-\frac{5}{2}\right)^2 - (-14)} \]
Calculating \(g^2\) and \(c\):
\[ g^2 = \left(-\frac{5}{2}\right)^2 = \frac{25}{4} \]
\[ c = -14 \implies -c = 14 \]
Now substituting these values:
\[ \text{Intercept on x-axis} = 2\sqrt{\frac{25}{4} + 14} = 2\sqrt{\frac{25}{4} + \frac{56}{4}} = 2\sqrt{\frac{81}{4}} = 2 \cdot \frac{9}{2} = 9 \]

### Step 3: Find the Y-axis Intercept
The formula for the y-axis intercept is:
\[ \text{Intercept on y-axis} = 2\sqrt{f^2 - c} \]
Substituting the values of \(f\) and \(c\):
\[ \text{Intercept on y-axis} = 2\sqrt{\left(-\frac{13}{2}\right)^2 - (-14)} \]
Calculating \(f^2\) and \(c\):
\[ f^2 = \left(-\frac{13}{2}\right)^2 = \frac{169}{4} \]
Now substituting these values:
\[ \text{Intercept on y-axis} = 2\sqrt{\frac{169}{4} + 14} = 2\sqrt{\frac{169}{4} + \frac{56}{4}} = 2\sqrt{\frac{225}{4}} = 2 \cdot \frac{15}{2} = 15 \]

### Final Result
Thus, the intercepts made by the circle on the x-axis and y-axis are:
\[ \text{Intercept on x-axis} = 9, \quad \text{Intercept on y-axis} = 15 \]
The answer is \( (9, 15) \).

---