Given the equation of conic C is $5x^2 + 6xy + 5y^2 = 8$, find the equation S of its auxiliary circle?
Now, I know that the equation C is an ellipse.
Since $\Delta = 5*5(-8) – (-8)(3^2) \neq 0$
And,
$ab – h^2 = 5*5 – 3^2 > 0$
Which is the condition for an ellipse.
But this isn't a standard one!. In my school, we have only worked with ellipses of the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
So, how do get the equation of the auxiliary circle for the given ellipse?
Any help would be appreciated.
Best Answer
Since the conic is centre origin, we may use polar coordinates:
\begin{align} 8 &= 5r^2\cos^2 \theta+6r^2\cos \theta \sin \theta+5r^2\sin^2 \theta \\ r^2 &= \frac{8}{5+6\cos \theta \sin \theta} \\ &= \frac{8}{5+3\sin 2\theta} \end{align}
Now, $$\frac{8}{5+3} \le r^2 \le \frac{8}{5-3} \implies 1 \le r^2 \le 4 \\$$