Question

Find the equation of the ellipse in the case: 

 eccentricity e = \[\frac{1}{2}\]  and semi-major axis = 4

 

Answer 1

\[ e = \frac{1}{2} \text{ and semi major axis } = 4\]
\[i . e . a = 4\]
\[\text{ We have }  e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow \frac{1}{2} = \sqrt{1 - \frac{b^2}{16}}\]
\[\text{ On squaring both sides, we get } :\]
\[\frac{1}{4} = \frac{16 - b^2}{16}\]
\[ \Rightarrow 16 = 64 - 4 b^2 \]
\[ \Rightarrow b^2 = 12\]
\[\text{ Substituting the values of }  a^2 \text{ and } b^2 \text{ ,we get } :\]
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
\[ \Rightarrow \frac{x^2}{16} + \frac{y^2}{12} = 1\]
\[ \Rightarrow \frac{3 x^2 + 4 y^2}{48} = 1\]
\[ \Rightarrow 3 x^2 + 4 y^2 = 48\]
\[\text{ This is the required equation of the ellipse } .\]