Question

Match List-I with List-II

\[ \begin{array}{|l|l|} \hline \textbf{List-I} & \textbf{List-II} \\ \hline (A) \; 9x^2 - 12xy + 4y^2 - 74x - 98y + 324 = 0 & (I) \; \text{Hyperbola} \\ (B) \; 12x^2 + 7xy - 12y^2 + 10x + 55y - 125 = 0 & (II) \; \text{A pair of straight lines} \\ (C) \; x^2 + 3xy + 2y^2 + x + y = 0 & (III) \; \text{Ellipse} \\ (D) \; 5x^2 + y^2 - 30x + 1 = 0 & (IV) \; \text{Parabola} \\ \hline \end{array} \]
Choose the correct answer from the options given below:

• (A) - (I), (B) - (III), (C) - (IV), (D) - (II)
• (A) - (IV), (B) - (I), (C) - (II), (D) - (III)
• (A) - (IV), (B) - (III), (C) - (II), (D) - (I)
• (A) - (IV), (B) - (I), (C) - (III), (D) - (II)

Hint:

The sign of \(h^2 - ab\) is the quickest way to classify a conic section when \(\Delta \neq 0\).
\(h^2-ab>0\): Hyperbola (think \(y^2-x^2=1\))
\(h^2-ab<0\): Ellipse (think \(x^2+y^2=1\))
\(h^2-ab = 0\): Parabola (think \(y=x^2\))
Always check for degeneracy (\(\Delta=0\)) first if the option of "pair of straight lines" is available.

Answer 1

The Correct Option is B

Step 1: Concept Identification
The conic section represented by \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) is determined by the discriminant \( \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 \) and the sign of \( h^2 - ab \).
If \( \Delta = 0 \), the equation represents a pair of straight lines.
If \( \Delta eq 0 \):
   If \( h^2 - ab<0 \), it's an Ellipse.
    If \( h^2 - ab = 0 \), it's a Parabola.
    If \( h^2 - ab>0 \), it's a Hyperbola.

Step 2: Detailed Analysis
    (A) \(9x^2 - 12xy + 4y^2 - 74x - 98y + 324 = 0\): \(a=9, h=-6, b=4\). Calculation: \(h^2 - ab = (-6)^2 - 9(4) = 36 - 36 = 0\). This signifies a Parabola. Thus, (A) corresponds to (IV).
    (B) \(12x^2 + 7xy - 12y^2 + 10x + 55y - 125 = 0\): \(a=12, h=7/2, b=-12\). Calculation: \(h^2 - ab = (7/2)^2 - 12(-12) = 49/4 + 144>0\). This signifies a Hyperbola. Thus, (B) corresponds to (I).
    (C) \(x^2 + 3xy + 2y^2 + x + y = 0\): \(a=1, h=3/2, b=2, g=1/2, f=1/2, c=0\). First, evaluate \( \Delta \): \( \Delta = (1)(2)(0) + 2(1/2)(1/2)(3/2) - 1(1/2)^2 - 2(1/2)^2 - 0 = 3/4 - 1/4 - 2/4 = 0 \). Since \( \Delta = 0 \), it represents a pair of straight lines. Thus, (C) corresponds to (II).
    (D) \(5x^2 + y^2 - 30x + 1 = 0\): \(a=5, h=0, b=1\). Calculation: \(h^2 - ab = 0^2 - 5(1) = -5<0\). This signifies an Ellipse. Thus, (D) corresponds to (III).

Step 3: Conclusion
The correct pairings are (A)-(IV), (B)-(I), (C)-(II), and (D)-(III). This aligns with option (B).