Question

Assertion :$C:(a{x}^2+b{y}^2+c)(a{y}^2-bx)=0$; $a$ and $b$ are nonzero real numbers.
$C$ represents a parabola and a pair of perpendicular lines if $c=0$ and $a$ and $b$ are of opposite signs. Reason:

$C:(a{x}^2+b{y}^2+c)(a{y}^2-bx)=0$; $a$ and $b$ are nonzero real numbers.

$C$ represents a parabola and a hyperbola if $a$ and $b$ are of opposite signs.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Given equation represents

$a{x}^2+b{y}^2+c=0,y^2=\frac{b}{a}x$

Now When $c=0,$ and a and b are of opposite sign $a{x}^2+b{y}^2=0$

represent a pair of straight lines and $y^2=\frac{b}{a}x$ represent a parabola..

Next, when a & b are of the opposite sign

$a{x}^2+b{y}^2=-c\Rightarrow \frac{x^2}{-c/a}-\frac{y^2}{c/b}=1$

$ab<0$, so it represents a hyperbola.

and $y^2=\frac{b}{a}x$ represent a parabola.

So both assertion and Reason are independently true.
Hence the correct choice is an option ${}^{\prime }{B}^{\prime }$