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: Conditions for a circle:
The coefficient of \(xy\) must be 0 and the coefficients of \(x^2\) and \(y^2\) must be equal.

Step 2: Find p and q:
\(3-p=0\) gives \(p=3\). Then \(q=3\).

Step 3: Rewrite the equation:
\(3x^2+3y^2-6x=8(3)(3)=72\). Divide by 3: \(x^2+y^2-2x=24\). Complete the square: \((x-1)^2+y^2=25\).

Step 4: Find the area:
Radius \(=5\), area \(=\pi r^2=25\pi\). Option C.

Step 5: Why the other options are wrong.
\(5\pi\), \(9\pi\), \(81\pi\) would need radii \(\sqrt5\), 3 and 9, which come from sign or arithmetic slips.

Final Answer:
The area is 25 pi. \[ \boxed{\text{(C) }25\pi} \]