Understanding the Condition for a Hyperbola

The given question asks for the condition under which the general equation of the second degree, $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$, represents a hyperbola. The type of conic section represented by this equation depends on the values of the coefficients $a, b,$ and $h$.

Identifying Conic Sections

The nature of the conic section represented by the general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ is primarily determined by two quantities: the discriminant $\Delta$ and the expression $h^2 - ab$.

The discriminant is given by:

$\Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = abc + 2fgh - af^2 - bg^2 - ch^2$

The expression $h^2 - ab$ relates to the quadratic terms $ax^2 + 2hxy + by^2$.

Conditions for Different Conic Sections

Assuming the equation represents a non-degenerate conic section (i.e., $\Delta \neq 0$), the type of conic is determined by the sign of $h^2 - ab$:

• If $h^2 - ab < 0$, the equation represents an Ellipse.
• If $h^2 - ab = 0$, the equation represents a Parabola.
• If $h^2 - ab > 0$, the equation represents a Hyperbola.

If $\Delta = 0$, the equation represents a pair of straight lines. The type of lines depends on $h^2 - ab$:

• If $h^2 - ab < 0$, the equation represents a pair of imaginary lines intersecting at a real point.
• If $h^2 - ab = 0$, the equation represents a pair of parallel lines (real and distinct, or coincident).
• If $h^2 - ab > 0$, the equation represents a pair of real and distinct intersecting lines.

Condition for a Hyperbola

The question asks for the condition for a hyperbola. For the general second-degree equation to represent a hyperbola, two main conditions must typically be met:

1. The conic must be non-degenerate, meaning $\Delta \neq 0$.
2. The condition on the coefficients of the quadratic terms must satisfy $h^2 - ab > 0$.

The options provided only relate to the condition involving $h^2$ and $ab$. The fundamental condition that distinguishes a hyperbola among non-degenerate conics is $h^2 - ab > 0$. This can be rewritten as $h^2 > ab$.

Analyzing the Options

Let's examine the given options based on the condition $h^2 > ab$ for a hyperbola:

• Option 1: $h^2 < ab$. This corresponds to $h^2 - ab < 0$, which represents an Ellipse (if non-degenerate).
• Option 2: $h^2 = ab$. This corresponds to $h^2 - ab = 0$, which represents a Parabola (if non-degenerate).
• Option 3: ${h^2} > \sqrt {ab}$. This is not the standard condition. The standard condition directly compares $h^2$ and $ab$.
• Option 4: $h^2 > ab$. This corresponds to $h^2 - ab > 0$, which represents a Hyperbola (if non-degenerate).

Therefore, the general equation of the second degree represents a hyperbola if the condition $h^2 > ab$ is met (assuming it's a non-degenerate case). This is the key condition derived from the classification of conic sections based on the discriminant and the $h^2 - ab$ term.

Revision Table: Conic Section Conditions

Additional Information: Degenerate vs. Non-degenerate Conics

It's important to remember the role of the discriminant $\Delta$.

• Non-degenerate Conics: When $\Delta \neq 0$, the equation represents an actual conic section (Ellipse, Parabola, or Hyperbola) with finite or infinite size, not reducing to lines or points. The type is determined by $h^2 - ab$.
• Degenerate Conics: When $\Delta = 0$, the equation represents a pair of straight lines, a single point, or no real locus. These are called degenerate conic sections and occur when the plane slicing the cone passes through its vertex. The value of $h^2 - ab$ further classifies the nature of these lines.

The question typically implies a non-degenerate conic when asking for the type like "hyperbola", hence focusing on the $h^2 - ab$ condition is appropriate, assuming $\Delta \neq 0$.