Question

The equation of a circle whose Centre is \((-3,2)\) and area is \(176\) units is

• A. \(x^2+y^2+6x-4y-36=0\)
• B. \(x^2+y^2+6x-4y-43=0\)
• C. \(x^2+y^2-6x+4y-36=0\)
• D. \(x^2+y^2-6x+4y-43=0\)

Hint:

For circle problems, first find \(r^2\) from the given area, then use \((x-h)^2+(y-k)^2=r^2\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find the general equation of a circle given its center coordinates and its area.
Step 2: Key Formula or Approach:
1. Area of circle \( A = \pi r^2 \). Use \( \pi \approx 22/7 \).
2. Standard equation of circle: \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center.
3. General form: \( x^2 + y^2 + 2gx + 2fy + c = 0 \).
Step 3: Detailed Explanation:

Step 3.1: Find the radius squared (\( r^2 \)):
Area = 176.
\( \frac{22}{7} \times r^2 = 176 \)
\( r^2 = 176 \times \frac{7}{22} \)
\( r^2 = 8 \times 7 = 56 \).

Step 3.2: Use the center-radius form:
Center \( (h, k) = (-3, 2) \).
Equation: \( (x - (-3))^2 + (y - 2)^2 = 56 \)
\( (x + 3)^2 + (y - 2)^2 = 56 \)

Step 3.3: Expand to general form:
\( (x^2 + 6x + 9) + (y^2 - 4y + 4) = 56 \)
\( x^2 + y^2 + 6x - 4y + 13 = 56 \)
\( x^2 + y^2 + 6x - 4y + 13 - 56 = 0 \)
\( x^2 + y^2 + 6x - 4y - 43 = 0 \).

Step 4: Final Answer:
The equation of the circle is \( x^2 + y^2 + 6x - 4y - 43 = 0 \).