Let $a$ and $b$ be non-zero real numbers. Then, the equation $$ \left(a x^{2}+b y^{2}+c\right)\left(x^{2}-5 x y+6 y^{2}\right)=0 $$ represents

(B) two straight lines and a circle, when $a=b$, and $c$ is of sign opposite to that of $a$

(A) four straight lines, when $c=0$ and $a, b$ are of the same sign

(C) two straight lines and a hyperbola, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$

(D) a circle and an ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$

JEE Advanced · 2008 · 13 Apr 2008 · Paper 1 · Straight Lines

Let

a

and

b

be non-zero real numbers. Then, the equation

(a x^{2} + b y^{2} + c) (x^{2} - 5 x y + 6 y^{2}) = 0

represents

(A)four straight lines, when $c = 0$ and $a, b$ are of the same sign
(B)two straight lines and a circle, when $a = b$ , and $c$ is of sign opposite to that of $a$ ✓ Correct
(C)two straight lines and a hyperbola, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
(D)a circle and an ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$

Answer: (B)

The equation factors as:

(a x^{2} + b y^{2} + c) (x^{2} - 5 x y + 6 y^{2}) = 0

The second factor

x^{2} - 5 x y + 6 y^{2} = (x - 2 y) (x - 3 y) = 0

represents two straight lines:

x = 2 y

and

x = 3 y

.

The first factor

a x^{2} + b y^{2} + c = 0

represents:

When $a = b$ and $c$ is of opposite sign to $a$ : $x^{2} + y^{2} = - \frac{c}{a} > 0$ , which is a circle.
When $a \neq = b$ (same sign) and $c$ is of opposite sign: an ellipse.
When $a$ and $b$ have opposite signs: a hyperbola.
When $c = 0$ : a point or imaginary.

Therefore, when

a = b

and

c

has opposite sign to

a

, the equation represents two straight lines and a circle.

JEE Advanced · 2008 · 13 Apr 2008 · Paper 1 · Single correct option