Question

The equation $12x^2-23xy+10y^2-25x+26y-14=0$ represents

• A. a parabola
• B. a circle
• C. a hyperbola
• D. an ellipse

Answer 1

The correct option is C a hyperbola

$12x^2-23xy+10y^2-25x+26y-14=0$
On comparing with $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0,$ we have
$a=12,h=-\frac{23}{2},b=10,g=-\frac{25}{2},f=13,c=-14$
$\Rightarrow h^2=\frac{529}{4},ab=120$
$\Rightarrow h^2>ab\cdots \left(i\right)$

Now, $△=abc+2fgh-a{f}^2-b{g}^2-c{h}^2$
$=12\left(10\right)(-14)+2\left(13\right)(-\frac{25}{2})(-\frac{23}{2})-12(13{)}^2-10{(-\frac{25}{2})}^2-(-14){(-\frac{23}{2})}^2$
$=\frac{637}{2}\ne 0\cdots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$, the above equation represents a hyperbola.