Question

What conics do the following equations represent? When possible, find their centres, and also their equations referred to the centre.
$12x^2-23xy+10y^2-25x+26y=14.$

Answer 1

The standard equation of any conic is written as $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$
When compared the given equation to this, we get
$a=12,h=-\frac{23}{2},b=10,g=\frac{-25}{2},f=13,c=-14$
$h^2=\frac{529}{4},ab=120$
$\therefore h^2>ab$
Also, $\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=\left|\begin{array}{ccc}12& \frac{-23}{2}& \frac{-25}{2}\\ \frac{-23}{2}& 10& 13\\ \frac{-25}{2}& 13& -14\end{array}\right|$
$=12\times (-140-169)+\frac{23}{2}\times (14\times \frac{23}{2}+13\times \frac{25}{2})-\frac{25}{2}\times (\frac{-23}{2}\times 13+10\times \frac{25}{2})$
$=-3708+\frac{23}{2}\times (161+\frac{325}{2})-\frac{25}{2}\times \left(\frac{250-299}{2}\right)$
$=-3708+\frac{14881}{4}+\frac{49\times 25}{4}$
$=-3708+\frac{14881+1225}{4}$
$=\frac{16106-14832}{4}$
$\ne 0$
When $h^2>ab$ and the determinant is not equal to zero, the conic is a hyperbola.
Differentiating the conic equation w.r.t $x$, we have
$24x-23y-25=0...\left(1\right)$
Differentiating the conic equation w.r.t $y$, we have
$-23x+20y+26=0...\left(2\right)$
Multiplying equation $\left(1\right)$ by $23$ and equation $\left(2\right)$ by $24$ and adding the two, we get
$-529y-575+480y+624=0$
i.e. $49y=49$ or $y=1$
Correspondingly, $24x-23-25=0$ or $x=2$
The center of the hyperbola is thus $(2,1)$