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

### Step 1: Rewrite the Circle Equation
The equation of the circle is given as:
\[ x^2 + y^2 + 4x - 7y + 12 = 0 \]
We can compare this with the general form of a circle:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From this, we can identify:
- \( 2g = 4 \) → \( g = 2 \)
- \( 2f = -7 \) → \( f = -\frac{7}{2} \)
- \( c = 12 \)

### Step 2: Use the Formula for Y-Intercept
The formula for the y-intercept of a circle is given by:
\[ \text{Intercept on y-axis} = 2\sqrt{f^2 - c} \]
Substituting the values of \( f \) and \( c \):
- \( f = -\frac{7}{2} \)
- \( c = 12 \)

### Step 3: Calculate \( f^2 \)
Calculate \( f^2 \):
\[ f^2 = \left(-\frac{7}{2}\right)^2 = \frac{49}{4} \]

### Step 4: Substitute into the Intercept Formula
Now substitute \( f^2 \) and \( c \) into the intercept formula:
\[ \text{Intercept} = 2\sqrt{\frac{49}{4} - 12} \]

### Step 5: Simplify the Expression
Convert \( 12 \) to a fraction with the same denominator:
\[ 12 = \frac{48}{4} \]
Now substitute:
\[ \text{Intercept} = 2\sqrt{\frac{49}{4} - \frac{48}{4}} = 2\sqrt{\frac{1}{4}} \]

### Step 6: Calculate the Final Value
Simplifying further:
\[ \text{Intercept} = 2 \cdot \frac{1}{2} = 1 \]

### Conclusion
Thus, the intercept on the y-axis is:
\[ \boxed{1} \]

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