Question

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

• A. four straight lines, when $c=0$ and $a,b$ are of the opposite sign
• B. two straight lines and a circle, when $a=b$ and $c$ is of sign opposite to that of $a$
• C. a circle and ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
• D. two straight lines and a circle,when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$

Answer 1

The correct option is B two straight lines and a circle, when $a=b$ and $c$ is of sign opposite to that of $a$

by rearranging the given equation we will get

$(a{x}^2+b{y}^2+c)(x-3y)(x-2y)=0$

$(x-3y)(x-2y)$ represemts the pair of straight lines
$a{x}^2+b{y}^2+c$ circle if $a=b,c=-a$