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

Options

A. 1
B. 3
C. 5
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 1: Rewrite the Circle Equation
The given equation of the circle is:
\[ x^2 + y^2 + 4x - 7y + 12 = 0 \]
We can rearrange it to match the standard form of a circle equation \( (x - h)^2 + (y - k)^2 = r^2 \).

### Step 2: Complete the Square
We will complete the square for both \( x \) and \( y \).

For \( x \):
\[ x^2 + 4x \quad \text{can be written as} \quad (x + 2)^2 - 4 \]

For \( y \):
\[ y^2 - 7y \quad \text{can be written as} \quad (y - \frac{7}{2})^2 - \frac{49}{4} \]

### Step 3: Substitute Back into the Equation
Substituting these back into the equation gives:
\[ (x + 2)^2 - 4 + (y - \frac{7}{2})^2 - \frac{49}{4} + 12 = 0 \]

Combining the constants:
\[ -4 - \frac{49}{4} + 12 = -4 - 12.25 + 12 = -4.25 \]
Thus, we have:
\[ (x + 2)^2 + (y - \frac{7}{2})^2 = \frac{49}{4} - 4.25 \]

### Step 4: Calculate the Right Side
Calculating the right side:
\[ \frac{49}{4} - 4.25 = \frac{49}{4} - \frac{17}{4} = \frac{32}{4} = 8 \]
So, the equation of the circle is:
\[ (x + 2)^2 + (y - \frac{7}{2})^2 = 8 \]

### Step 5: Find the y-axis Intercept
The y-axis intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the circle equation:
\[ (0 + 2)^2 + (y - \frac{7}{2})^2 = 8 \]
This simplifies to:
\[ 4 + (y - \frac{7}{2})^2 = 8 \]
\[ (y - \frac{7}{2})^2 = 4 \]

### Step 6: Solve for y
Taking the square root of both sides gives:
\[ y - \frac{7}{2} = 2 \quad \text{or} \quad y - \frac{7}{2} = -2 \]
Thus,
\[ y = \frac{7}{2} + 2 = \frac{11}{2} \quad \text{or} \quad y = \frac{7}{2} - 2 = \frac{3}{2} \]

### Step 7: Calculate the Length of the Intercept
The y-axis intercepts are \( \frac{11}{2} \) and \( \frac{3}{2} \). The length of the intercept on the y-axis is:
\[ \frac{11}{2} - \frac{3}{2} = \frac{8}{2} = 4 \]

### Conclusion
The length of the intercept on the y-axis is \( 4 \).