Step 1
Formula Used:
Eccentricity PSK(e)=1\ -\ b2a2
Step 2
Calculation:
Compare the given equation with the equation of an ellipse \(\displaystyle{x}{2}{a}{2}\ +\ {y}{2}{b}{2}={1}\) and we get:
\(\displaystyle{a}{2}={9}\ \Rightarrow\ {a}={3}{b}{2}={4}\ \Rightarrow\ {b}={2}\)
Now,
\(\displaystyle{e}={1}\ -\ {b}{2}{a}{2}={1}\ -\ {49}={53}\)
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 vertice of the conic section and we get:
\(\displaystyle{\left(\pm\ {a},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {b}\right)}={\left(\pm\ {3},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {2}\right)}\)
Now,
\(\displaystyle\text{Foci}\ ={\left(\pm\ {a}{e},\ {0}\right)}={\left(\pm\ {5},\ {0}\right)}\)
Thus, the vertices and foci of the conic section are: \(\displaystyle{\left(\pm\ {3},\ {0}\right)}\ \text{and}\ {\left({0},\ \pm\ {2}\right)}\ \text{and}\ {\left(\pm\ {5},\ {0}\right)}.\)