91Ó°ÊÓ

In Exercises 1 through 6 , find the eccentricity, center, foci, and directrices of each of the given ellipses and draw a sketch of the graph. $$ 6 x^{2}+9 y^{2}-24 x-54 y+51=0 $$

For each of the parabolas in Exercises 1 through 8 , find the coordinates of the focus, an equation of the directrix, and the length of the latus rectum. Draw a sketch of the curve. $$ x^{2}=4 y $$

In Exercises 1 through 4 , find the vertices, foci, directrices, eccentricity, and ends of the minor axis of the given ellipse. Draw a sketch of the curve and show the foci and the directrices. $$ 4 x^{2}+9 y^{2}=36 $$

Show that an equation of a conic having its principal axis along the polar axis and its extension, a focus at the pole, and the corresponding directrix to the right of the focus is \(r=e d /(1+e \cos \theta)\).

For each of the parabolas in Exercises 1 through 8 , find the coordinates of the focus, an equation of the directrix, and the length of the latus rectum. Draw a sketch of the curve. $$ y^{2}=6 x $$

Find the vertices, foci, directrices, eccentricity, and ends of the minor axis of the given ellipse. Draw a sketch of the curve and show the foci and the directrices. $$ 4 x^{2}+9 y^{2}=4 $$

Find the eccentricity, center, foci, and directrices of each of the given ellipses and draw a sketch of the graph. $$ 9 x^{2}+4 y^{2}-18 x+16 y-11=0 $$

Show that an equation of a conic having its principal axis along the \(\frac{1}{2} \pi\) axis and its extension, a focus at the pole, and the corresponding directrix above the focus is \(r=e d /(1+e \sin \theta)\).

In Exercises 2 through 8, remove the \(x y\) term from the given equation by a rotation of axes. Draw a sketch of the graph and show both sets of axes. $$ x^{2}+x y+y^{2}=3 $$

Find the vertices, foci, directrices, eccentricity, and ends of the minor axis of the given ellipse. Draw a sketch of the curve and show the foci and the directrices. $$ 2 x^{2}+3 y^{2}=18 $$