Question

The equation $2x^2-3xy+5y^2+6x-3y+5=0$ represents

• A. a parabola
• B. an ellipse
• C. a hyperbola
• D. a pair of straight lines

Answer 1

Answer: (B)

an ellipse

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

$a=2,h=-\frac{3}{2},b=5,g=3,f=-\frac{3}{2},c=5$

$\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2$

$=\left(2\right)\left(5\right)\left(5\right)+2(-\frac{3}{2})\left(3\right)(-\frac{3}{2})-\left(2\right){(-\frac{3}{2})}^2-\left(5\right)(3{)}^2-(5){(-\frac{3}{2})}^2$

$=50+\frac{27}{2}-\frac{9}{2}-45-\frac{225}{4}$

$=-\frac{169}{4}\ne 0$

Descriminant $=h^2-ab$

$=(-\frac{3}{2}{)}^2-(2\left)\right(5)$

$=\frac{9}{4}-10$

$=\frac{-31}{4}<0$

So, the curve represents either a circle or an ellipse

$a\ne b$ and $\frac{\Delta }{a+b}=\frac{-\frac{169}{4}}{2+5}=-\frac{169}{28}<0$

So, the curve represents a ellipse.

So, the answer is option (B).