# 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#?

##### 1 Answer

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

So we can factor by completing the square:

or in a more familiar form,

**Vertices **

This means that the major axis is in the

**Foci**

We see the two axial lengths:

so the foci are at

**End Points of the Minor Axis**

This means that the minor axis is in the

**Center **

The center is just

**Latus Recta **

The latus recta go through the foci and intersect the graph. We know that the

where the two

**Directricies **

The directrices will be vertical lines a distance of

**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.