Question

Let $a$ and $b$ be 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 same sign
• B. two straight lines and a circle, when $a=b$, and $c$ is of sign opposite to that of $a$
• 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 1

Answer: (B)

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

$(a{x}^2+b{y}^2+c)(x^2-5xy+6y^2)=0$
$\Rightarrow a{x}^2+b{y}^2+c=0$ or $x^2-5xy+6y^2=0$
$\Rightarrow x^2+y^2=(-\frac{c}{a})$ if $a=b,x-2y=0$ and $x-3y=0$
Hence the given equation represents two straight lines and a circle,

when $a=b$ and $c$ is of sign opposite to that of $a$.