Question

The equation $7y^2-9x^2+54x-28y-116=0$ represents

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

Answer 1

The correct option is A a hyperbola

Given conic is $7y^2-9x^2+54x-28y-116=0$

Here $a=-9,b=7,c=-116,f=-14,g=27,h=0$

$\therefore \Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2=-\mathrm{9.7.}(-116)+0-9(14{)}^2-7(27{)}^2-0=441\ne 0$

and $h^2=0,ab=-63$

clearly $h^2>ab$

we know that for a equation $a{x}^2+b{y}^2+2hxy2gx+2fy+c=0$ it represents a hyperbola if $\Delta \ne 0$ and $h²>ab$

Hence given conic represent hyperbola.