Question

If the equation \(3x^2+(3-p)xy+qy^2-2px = 8pq\) represents a circle, then the area (in sq. units) of this circle is

• A. \(5π\)
• B. \(9π\)
• C. \(25π\)
• D. \(81π\)

Hint:

For a circle, the xy term vanishes and the coefficients of x^2 and y^2 are equal.

Answer 1

The Correct Option is C

Step 1: Match with the general form:
The form is $Ax^2+2Hxy+By^2+2Gx+2Fy+C=0$ and a circle needs $H=0$, $A=B$.

Step 2: Solve:
$p=3,\ q=3$. Equation: $3x^2+3y^2-6x-72=0$, so $g=-1,\ f=0,\ c=-24$ after dividing by 3.

Step 3: Radius:
$r=\sqrt{g^2+f^2-c}=\sqrt{1+24}=5$. Area $=25\pi$, option C.

Final Answer:
With p = q = 3, the radius is 5, so the area is 25 pi. \[ \boxed{\text{(C) }25\pi} \]