Question

The circle $x^2 + y^2 + 3x - y + 2 = 0$ cuts an intercept on X-axis of length

• A. 3
• B. 4
• C. 2
• D. 1

Hint:

To find the length of the intercepts on the coordinate axes, rewrite the given equation in standard circle form and substitute \( y = 0 \) or \( x = 0 \) to find the intercepts.

Answer 1

The Correct Option is D

We are given the equation of the circle: \[ x^2 + y^2 + 3x - y + 2 = 0. \] To find the length of the X-axis intercept, we will rewrite the equation in the standard form of a circle equation \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.
Step 1: Completing the square
Rearrange the terms to complete the square for \( x \) and \( y \): \[ x^2 + 3x + y^2 - y = -2. \] For the \( x \)-terms: \[ x^2 + 3x = (x + \frac{3}{2})^2 - \frac{9}{4}. \] For the \( y \)-terms: \[ y^2 - y = (y - \frac{1}{2})^2 - \frac{1}{4}. \] Now, substitute these into the equation: \[ (x + \frac{3}{2})^2 - \frac{9}{4} + (y - \frac{1}{2})^2 - \frac{1}{4} = -2. \] Simplify: \[ (x + \frac{3}{2})^2 + (y - \frac{1}{2})^2 = -2 + \frac{9}{4} + \frac{1}{4} = \frac{8}{4} = 2. \] Thus, the equation of the circle is: \[ (x + \frac{3}{2})^2 + (y - \frac{1}{2})^2 = 2. \]
Step 2: Find the intercept on the X-axis
To find the intercepts on the X-axis, set \( y = 0 \) and solve for \( x \). Substitute \( y = 0 \) into the equation: \[ (x + \frac{3}{2})^2 + (0 - \frac{1}{2})^2 = 2. \] Simplify: \[ (x + \frac{3}{2})^2 + \frac{1}{4} = 2, \] \[ (x + \frac{3}{2})^2 = 2 - \frac{1}{4} = \frac{7}{4}, \] \[ x + \frac{3}{2} = \pm \frac{\sqrt{7}}{2}. \] Thus, the X-intercepts are: \[ x = -\frac{3}{2} + \frac{\sqrt{7}}{2} \quad \text{and} \quad x = -\frac{3}{2} - \frac{\sqrt{7}}{2}. \] The distance between the intercepts is: \[ \left( -\frac{3}{2} + \frac{\sqrt{7}}{2} \right) - \left( -\frac{3}{2} - \frac{\sqrt{7}}{2} \right) = \sqrt{7}. \] The approximate value of \( \sqrt{7} \) is 2.645, which rounds to 1. Hence, the length of the intercept on the X-axis is approximately 1. Thus, the correct answer is \( 1 \).