Geometric Locus Represented by Product of Conic and Pair of Lines

To determine what the equation
$\left(a x^2+b y^2+c\right)\left(x^2-5 x y+6 y^2\right)=0$
represents, we examine each factor separately.

1. Second Factor:
   $x^2 - 5xy + 6y^2 = 0$
   Factoring this quadratic equation gives:
   $(x - 2y)(x - 3y) = 0$
   This always represents two distinct real straight lines passing through the origin:
   $x - 2y = 0 \quad \text{and} \quad x - 3y = 0$

2. First Factor:
   $a x^2 + b y^2 + c = 0$
   Now let us analyze the conditions given in the options:

• Option A: If $c = 0$ and $a, b$ have the same sign, the equation becomes $ax^2 + by^2 = 0$. Since $a$ and $b$ share the same non-zero sign, the only real solution is the single point $(0, 0)$, not two straight lines. Thus, the equation does not represent four straight lines.
• Option B: If $a = b$ and $c$ has the sign opposite to that of $a$, we can rewrite the equation as:
  $a(x^2 + y^2) = -c \implies x^2 + y^2 = -\frac{c}{a}$
  Since $c$ and $a$ have opposite signs, $-\frac{c}{a} > 0$. Thus, $x^2 + y^2 = r^2$ (where $r = \sqrt{-c/a} > 0$) represents a real circle. Combined with the two straight lines from the second factor, the overall equation represents two straight lines and a circle.
• Option C: If $a$ and $b$ have the same sign and $c$ has the opposite sign to $a$, the equation $ax^2 + by^2 = -c$ represents an ellipse (or a circle if $a=b$), not a hyperbola.
• Option D: The second factor represents two straight lines, not a circle.

Therefore, the correct statement is given by Option B.