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

Concept: The equation of a circle with centre \((h,k)\) and radius \(r\) is: \[ (x-h)^2+(y-k)^2=r^2 \]

Step 1: Given centre is: \[ (h,k)=(-3,2) \] So the equation becomes: \[ (x+3)^2+(y-2)^2=r^2 \]

Step 2: Area of circle is: \[ \pi r^2=176 \] Taking \(\pi=\frac{22}{7}\): \[ \frac{22}{7}r^2=176 \] \[ r^2=\frac{176\times 7}{22} \] \[ r^2=56 \]

Step 3: Substitute \(r^2=56\). \[ (x+3)^2+(y-2)^2=56 \]

Step 4: Expand. \[ x^2+6x+9+y^2-4y+4=56 \] \[ x^2+y^2+6x-4y+13-56=0 \] \[ x^2+y^2+6x-4y-43=0 \] Therefore, \[ \boxed{x^2+y^2+6x-4y-43=0} \]