6. For each of the following ellipses, find the coordinates of the center, vertices, and foci; the lengths of the major and minor axes; the eccentricity; and the equations of the directrices. Sketch the curve.

(a) [tex]$\frac{x^2}{16}+\frac{y^2}{4}=1$[/tex]

(b) [tex]$25x^2 + 9y^2 = 225$[/tex]

(c) [tex]$x^2 + 9y^2 + 4x - 18y - 23 = 0$[/tex]

7. Find the equation of the ellipse, given:

(a) Vertices [tex]$(\pm 8, 0)$[/tex], minor axis [tex]$= 6$[/tex]

(b) One vertex at [tex]$(0, 13)$[/tex], one focus at [tex]$(0, -12)$[/tex], and center at [tex]$(0, 0)$[/tex]

(c) Foci [tex]$(\pm 10, 0)$[/tex], eccentricity [tex]$=\frac{5}{6}$[/tex]

(d) Vertices [tex]$(8, 3)$[/tex] and [tex]$(-4, 3)$[/tex], one focus at [tex]$(6, 3)$[/tex]

(e) Directrices [tex]$4y - 33 = 0$[/tex], [tex]$4y + 17 = 0$[/tex]; major axis on [tex]$x + 1 = 0$[/tex]; eccentricity [tex]$=\frac{4}{2}$[/tex]