Question

The circle \( x^2 + y^2 + 4x - 7y + 12 = 0 \) cuts an intercept on the Y-axis of length

• A. 3
• B. 4
• C. 7
• D. 1

Hint:

To find the intercept of a circle on the coordinate axes, substitute \( x = 0 \) for the Y-axis intercept and \( y = 0 \) for the X-axis intercept in the equation of the circle.

Answer 1

The Correct Option is D

To find the Y-intercept, we set \( x = 0 \) in the equation of the circle. Substituting \( x = 0 \) into \( x^2 + y^2 + 4x - 7y + 12 = 0 \), we get: \[ y^2 - 7y + 12 = 0 \] This is a quadratic equation in \( y \). Solving using the quadratic formula: \[ y = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(12)}}{2(1)} \] \[ y = \frac{7 \pm \sqrt{49 - 48}}{2} = \frac{7 \pm \sqrt{1}}{2} \] \[ y = \frac{7 \pm 1}{2} \] Thus, the two values of \( y \) are: \[ y = 4 \quad \text{and} \quad y = 3 \] The length of the intercept is the difference between these two values: \[ \text{Intercept length} = 4 - 3 = 1 \] Therefore, the correct answer is 1.