Find the coordinate of the vertices foci, end points of the minor axis, center, latus recta, the equation of the directrices, sketch the equation of x^2+9y^2-24x-54y+51=0?

May 3, 2018

We know that this equation defines an ellipse, since it has the squared terms added together for both. Similarly, we know it is in line with the axes since there is no cross term of $x \mathmr{and} y$.

So we can factor by completing the square:
$\left({x}^{2} - 24 x + \textcolor{R E D}{144}\right) + 9 \left({y}^{2} - 6 y + \textcolor{R E D}{9}\right) \textcolor{B L U E}{- 144 - 81} + 51 = 0$

${\left(x - 12\right)}^{2} + 9 {\left(y - 3\right)}^{2} - 174 = 0$
or in a more familiar form,
${\left(x - 12\right)}^{2} / \left(174\right) + {\left(y - 3\right)}^{2} / \left(\frac{174}{9}\right) = 1$

Vertices
This means that the major axis is in the $x$ direction and has distance $\sqrt{174}$ from the center, i.e. the end points of the minor axis are $\left(12 - \sqrt{174} , 3\right) , \left(12 + \sqrt{174} , 3\right)$

Foci
We see the two axial lengths: $b = \sqrt{174}$ and $a = \frac{\sqrt{174}}{3}$ and the length to the focii is
${c}^{2} = {b}^{2} - {a}^{2} = 174 \left(1 - \frac{1}{9}\right) \rightarrow c = \frac{2 \sqrt{348}}{3}$
so the foci are at
$\left(12 - \frac{2 \sqrt{348}}{3} , 3\right) , \left(12 + \frac{2 \sqrt{348}}{3} , 3\right)$

End Points of the Minor Axis
This means that the minor axis is in the $y$ direction and has distance $\frac{\sqrt{174}}{3}$ from the center, i.e. the end points of the minor axis are $\left(12 , 3 - \frac{\sqrt{174}}{3}\right) , \left(12 , 3 + \frac{\sqrt{174}}{3}\right)$

Center
The center is just $\left(12 , 3\right)$.

Latus Recta
The latus recta go through the foci and intersect the graph. We know that the $x$ coordinates are $12 \pm c$, and we can either "plug and chug" or use the fact that we know a formula for the length of the latus rectum $\left(\frac{2 {a}^{2}}{b}\right)$ to find that the points are
$\left(12 \textcolor{R E D}{\pm} c , 3 \textcolor{B L U E}{\pm} {a}^{2} / b\right)$
where the two $\pm$ have nothing to do with each other, so we get 4 points.

Directricies
The directrices will be vertical lines a distance of ${b}^{2} / c$ from the center, i.e. $x = 12 \pm {b}^{2} / c$.

Sketch
From all this, we can very clearly sketch the equation:

where the vertices are green, the foci are magenta, the minor axis is red, the center is black, the latus recta are blue, and the directricies are a muted cyan.