Step 1
Formula: eccentricity \(\displaystyle{\left({e}\right)}={1}\ -\ {b}{2}{a}{2}\)
Step 2
Calculation:
Compare this equation with the standard ellipse equation \(\displaystyle{x}{2}{a}{2}\ +\ {y}{2}{b}{2}={1}\) and we get:
\(\displaystyle{a}{2}={81}\ \Rightarrow\ {a}={9}{b}{2}={16}\ \Rightarrow\ {b}={4}\)
Now,
\(\displaystyle{e}={1}\ -\ {b}{2}{a}{2}={1}\ -\ {1681}={659}\)
Vertices are:
\(\displaystyle{\left(\pm\ {a},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {b}\right)}\)
Now, put the values of a and b to get the vertices of the conic section and we get:
\(\displaystyle{\left(\pm\ {a},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {b}\right)}={\left(\pm\ {9},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {4}\right)}\)
Now,
\(\displaystyle\text{Foci}\ ={\left(\pm\ {a}{e},\ {0}\right)}={\left(\pm\ {65},\ {0}\right)}\)
Thus, the vertices and foci of the conic section are: \(\displaystyle{\left(\pm\ {9},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {4}\right)}\ \text{and}\ {\left(\pm\ {65},\ {0}\right)}.\)