Question

The equation$2y^2-xy-x^2+6x-8=0$ represents

• A. A pair of straight lines
• B. A circle
• C. An ellipse
• D. A parabola

Answer 1

The correct option is C

An ellipse

Identifying the curve:

Given $2y^2-xy-x^2+6x-8=0$

The standard form of the equation is $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

Comparing the above equation with the given equation, we get

$a=-1,h=-\frac{1}{2},b=2,g=3,f=0,c=-8$

We know that the condition for straight lines is given by
$\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2$

Substituting the values, we get

$$\begin{gathered} =(-1)\left(2\right)(-8)+2\left(0\right)\left(3\right)\left(\frac{-1}{2}\right)-(-1){\left(0\right)}^2-\left(2\right){\left(3\right)}^2-(-8){\left(\frac{-1}{2}\right)}^2 \\ =16+0+0-18+2 \\ \Rightarrow 0=0 \end{gathered}$$

Discriminant $=b^2-4ac$

$$\begin{gathered} ={(4)}^2-4(-1)(-8) \\ =16-32 \\ =-16<0 \end{gathered}$$

We know that if $b^2-4ac<0$, then the curve represents either a circle or an ellipse.

Now from the values, we know $a\ne c$.

For a curve to be an ellipse, $b^2-4ac<0$ and $a\ne c$. This proves that the given curve is an ellipse.

Hence, option (C) is the correct answer.