Answer
$$\eqalign{
& {\text{Center}}\left( {0,0} \right) \cr
& {\text{Foci }}\left( { - 3\sqrt 2 ,0} \right){\text{ and }}\left( {3\sqrt 2 ,0} \right) \cr
& {\text{Vertex }}\left( { - \sqrt {21} ,0} \right){\text{ and }}\left( {\sqrt {21} ,0} \right) \cr
& {\text{Eccentricity }}\frac{{2\sqrt 7 }}{7} \cr} $$
Work Step by Step
$$\eqalign{
& 3{x^2} + 7{y^2} = 63 \cr
& {\text{Divide both sides of the equation by }}63 \cr
& \frac{{3{x^2}}}{{63}} + \frac{{7{y^2}}}{{63}} = \frac{{63}}{{63}} \cr
& \frac{{{x^2}}}{{21}} + \frac{{{y^2}}}{9} = 1 \cr
& {\text{The equation has the standard form }}\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,{\text{ }}a > b \cr
& \frac{{{x^2}}}{{\left( {\sqrt {21} } \right)}} + \frac{{{y^2}}}{{{{\left( 3 \right)}^2}}} = 1 \Rightarrow a = \sqrt {21} ,b = 3 \cr
& c = \sqrt {{a^2} - {b^2}} = \sqrt {21 - 9} = 2\sqrt 3 \cr
& {\text{With}} \cr
& {\text{Vertex }}\left( { - a,0} \right){\text{ and }}\left( {a,0} \right) \cr
& {\text{Vertex }}\left( { - \sqrt {21} ,0} \right){\text{ and }}\left( {\sqrt {21} ,0} \right) \cr
& {\text{Center}}\left( {0,0} \right) \cr
& {\text{Foci }}\left( { - c,0} \right){\text{ and }}\left( {c,0} \right) \cr
& {\text{Foci }}\left( { - 3\sqrt 2 ,0} \right){\text{ and }}\left( {3\sqrt 2 ,0} \right) \cr
& {\text{Eccentricity }}e = \frac{c}{a} = \frac{{2\sqrt 3 }}{{\sqrt {21} }} = \frac{{2\sqrt 7 }}{7} \cr
& {\text{Graph:}} \cr} $$
