The circle `x^(2)+y^(2)+ 4x - 7y+12=0` cuts an intercept on y-axis equal to

Options

A. 1
B. 3
C. 4
D. 7

Answer

To find the y-axis intercept of the circle given by the equation \( x^2 + y^2 + 4x - 7y + 12 = 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Circle Equation**:
The given equation of the circle is:
\[ x^2 + y^2 + 4x - 7y + 12 = 0 \]

2. **Rewrite in Standard Form**:
We can rearrange this equation to identify the coefficients. The general form of a circle is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Here, we have:
\[ g = 2 \quad (since \, 2g = 4 \Rightarrow g = 2) \]
\[ f = -\frac{7}{2} \quad (since \, 2f = -7 \Rightarrow f = -\frac{7}{2}) \]
\[ c = 12 \]

3. **Use the Formula for Y-Intercept**:
The formula for the y-intercept of a circle is given by:
\[ \text{Intercept on y-axis} = 2 \sqrt{f^2 - c} \]

4. **Calculate \( f^2 \) and \( f^2 - c \)**:
First, we calculate \( f^2 \):
\[ f^2 = \left(-\frac{7}{2}\right)^2 = \frac{49}{4} \]
Now, calculate \( f^2 - c \):
\[ f^2 - c = \frac{49}{4} - 12 = \frac{49}{4} - \frac{48}{4} = \frac{1}{4} \]

5. **Find the Y-Intercept**:
Now substitute \( f^2 - c \) into the intercept formula:
\[ \text{Intercept on y-axis} = 2 \sqrt{\frac{1}{4}} = 2 \times \frac{1}{2} = 1 \]

### Final Answer:
The circle cuts an intercept on the y-axis equal to **1**.

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