Question

Let $a,b$ be non-zero real numbers. The equation $(a{x}^2+b{y}^2+c)(x^2-5xy+6y^2)$ 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

Answer 1

Answer: (B)

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

From given expression, $x^2-5xy+6y^2=0$ are pair of straight lines $y=\frac{x}{2}$ and $y=\frac{x}{3}$
Now, $a{x}^2+b{y}^2+c=0$ will be cirlce of radius $\sqrt{-\frac{c}{a}}$.

If $a=b$ and $c$ is of opposite sign of $a$ and $b$, then $x^2+y^2=-\frac{c}{a}$