Question

Let the equation \[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \] represent a point circle (not at the origin). Then which one of the following conditions must hold?

• A. \( b>0, c>0 \)
• B. \( b>0, c<0 \)
• C. \( b<0, c>0 \)
• D. \( b \leq 0, c<0 \)

Hint:

A point circle implies a single point; all terms should reduce to a perfect square with a zero radius.

Answer 1

The Correct Option is A

For the given second-degree general equation to represent a point circle: - The equation must be reducible to a perfect square. - The determinant condition for a circle to reduce to a point circle: \[ \text{Discriminant} \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \] Also, the radius = 0 condition implies that center exists and the square of radius = 0 ⇒ requires all coefficients be positive to allow real points. Thus, both \( b>0 \) and \( c>0 \) must hold.