Question

Let $C$ be the curve(s) given by $(a{x}^2+b{y}^2+c)(a{y}^2-bx)=0;a,b,c\in R-\left\{0\right\}$ and $a\ne b.$ Then $C$ represents

• A. a parabola only if $a,b$ and $c$ are of the same sign.
• B. a circle or a parabola if $a$ and $b$ are of the same sign and $c$ is of opposite sign.
• C. an ellipse or a parabola if $a$ and $b$ are of the same sign and $c$ is of opposite sign.
• D. a hyperbola or a parabola if $a$ and $b$ are of opposite sign

Answer 1

Answer: (A), (C), (D)

a hyperbola or a parabola if $a$ and $b$ are of opposite sign

$(a{x}^2+b{y}^2+c)(a{y}^2-bx)=0$
$a{x}^2+b{y}^2+c=0$ or $a{y}^2-bx=0$
Here $a{y}^2-bx=0$ represents a parabola $\forall a,b\in R-\left\{0\right\}$

$\left(i\right)$ For $a{x}^2+b{y}^2+c=0$
If $a,b,c$ are of the same sign, then $a{x}^2+b{y}^2+c=0$ is not possible. $(\because a,b,c\in R-\{0\left\}\right)$
So, $C$ will represent parabola only

$\left(ii\right)$ For $a{x}^2+b{y}^2=-c,$
$\Rightarrow \frac{x^2}{(-c/a)}+\frac{y^2}{(-c/b)}=1$
For the above equation to represent an Ellipse,
$\left(\frac{-c}{a}\right)>0,\left(\frac{-c}{b}\right)>0$
$\Rightarrow \left(\frac{c}{a}\right)<0,\left(\frac{c}{b}\right)<0$
$\Rightarrow$ $a$ and $b$ should be of the same sign but $c$ should be of opposite sign.

$\left(iii\right)$For $a{x}^2+b{y}^2=-c$ to represent hyperbola, $\left(\frac{-c}{a}\right)\left(\frac{-c}{b}\right)<0$
$\Rightarrow ab<0$
$\Rightarrow a$ and $b$ should be of the opposite sign