Question

The center of the circle \(4x^2 + 4y^2 - 8x + 12y - 25 = 0\) is:

• A. \((-2, 3)\)
• B. \((1, -3/2)\)
• C. \((-4, 6)\)
• D. \((4, -6)\)

Hint:

To quickly find the center of any circle equation:
1. Ensure the coefficients of \(x^2\) and \(y^2\) are equal to 1.
2. The center is simply:
\[ \left( -\frac{\text{coefficient of } x}{2}, -\frac{\text{coefficient of } y}{2} \right) \]
Here: \(\left( -\frac{-2}{2}, -\frac{3}{2} \right) = \left(1, -\frac{3}{2}\right)\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The general equation of a circle is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
The center of a circle written in this standard form is given by the coordinate:
\[ \text{Center} = (-g, -f) \]

Step 3: Detailed Explanation:
We are given the equation of the circle:
\[ 4x^2 + 4y^2 - 8x + 12y - 25 = 0 \]
First, we must divide the entire equation by 4 to make the coefficients of \(x^2\) and \(y^2\) equal to 1, which puts it into the standard form:
\[ x^2 + y^2 - 2x + 3y - \frac{25}{4} = 0 \]
Now, compare this with the general circle equation:
- The coefficient of \(x\) is \(2g = -2 \implies g = -1\).
- The coefficient of \(y\) is \(2f = 3 \implies f = \frac{3}{2}\).
Using these values, we find the coordinates of the center of the circle:
\[ \text{Center} = (-g, -f) \]
\[ \text{Center} = \left(-(-1), -\frac{3}{2}\right) = \left(1, -\frac{3}{2}\right) \]
This coordinate, \((1, -3/2)\), matches the second option.

Step 4: Final Answer:
Therefore, the correct option is (B).