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

To find the length of the X-axis intercept for the circle given by the equation:

\[ x^2 + y^2 + 3x - y + 2 = 0, \]

we follow these steps:

1. Compare with Standard Form:
The standard form of a circle's equation is:

\[ x^2 + y^2 + 2gx + 2fy + c = 0. \]
Comparing coefficients, we find:

\[ 2g = 3 \Rightarrow g = \frac{3}{2}, \quad 2f = -1 \Rightarrow f = -\frac{1}{2}, \quad c = 2. \]

2. X-Intercept Formula:
The length of the X-axis intercept is given by:

\[ 2\sqrt{g^2 - c}. \]

3. Substitute Values:
Plugging in the values for \( g \) and \( c \):

\[ 2\sqrt{\left(\frac{3}{2}\right)^2 - 2} = 2\sqrt{\frac{9}{4} - 2}. \]

4. Simplify the Expression:
Convert 2 to fourths and subtract:

\[ \frac{9}{4} - \frac{8}{4} = \frac{1}{4}. \]
Now take the square root:

\[ 2\sqrt{\frac{1}{4}} = 2 \cdot \frac{1}{2} = 1. \]

Final Answer:
The length of the X-axis intercept is \( 1 \).

Answer 2

The equation of the circle is \(x^2 + y^2 + 3x - y + 2 = 0\). 

Comparing it with the standard form \(x^2 + y^2 + 2gx + 2fy + c = 0\), we have \(g = \frac{3}{2}\), \(f = -\frac{1}{2}\), and \(c = 2\).

The x-intercept length is given by \(2\sqrt{g^2 - c}\). 

Substituting the values:

\(2\sqrt{\left(\frac{3}{2}\right)^2 - 2} = 2\sqrt{\frac{9}{4} - 2} = 2\sqrt{\frac{9-8}{4}} = 2\sqrt{\frac{1}{4}} = 2 \cdot \frac{1}{2} = 1\)

Therefore, the x-intercept length is 1.