Step-by-step Solution:

We are given the general second-degree equation of a conic: \[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \] where \( a, b, c, f, g, h \) are constants. Step 1: Condition for a Circle A conic represents a circle if: 1. The equation does not contain an \( xy \)-term, i.e., \( h = 0 \). 2. The coefficients of \( x^2 \) and \( y^2 \) are equal, i.e., \( a = b \neq 0 \). Now, let's analyze the given statements about circles: \[Statement (A):\] "The given conic is a circle if \( a = 0 \) and \( b = 0 \)." - If \( a = 0 \) and \( b = 0 \), the equation lacks any quadratic terms \( x^2 \) or \( y^2 \), meaning it does not represent a circle but rather a degenerate conic (a line or no curve at all). - Thus, Statement (A) is false. \[Statement (B):\] "The given conic is a circle if \( a = b \neq 0 \) and \( h = 0 \)." - This correctly satisfies the conditions for a circle (equal coefficients of \( x^2 \) and \( y^2 \), and no \( xy \)-term). - Thus, Statement (B) is true. \[Statement (C)\]: "The given conic is a circle if \( a = b \neq 0 \) and \( h \neq 0 \)." - If \( h \neq 0 \), the equation contains an \( xy \)-term, which means it represents a rotated conic, not a standard circle. - Thus, Statement (C) is false. Step 2: Condition for a Pair of Real and Distinct Straight Lines A second-degree equation represents a pair of real and distinct straight lines if: \[ f = g = c = 0 \quad \text{and} \quad h^2 - ab > 0. \] \[Statement (D)\]: This exactly matches the given condition. - Thus, Statement (D) is true. Step 3: Conclusion True statements: (B) and (D). The correct option is (B) and (D) only. Final Answer: \[ \boxed{\text{ (B) and (D) only.}} \]