Question

Find the equation of the ellipse in the case:

eccentricity e = \[\frac{1}{2}\]  and major axis = 12

 

 

Answer 1

\[ e = \frac{1}{2}\text{  and major axis } = 12\]
\[i . e . , 2a = 12 \text{ or } a = 6\]
\[\text{ We have e } = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow \frac{1}{2} = \sqrt{1 - \frac{b^2}{36}}\]
\[\text{ On squaring both sides, we get } :\]
\[\frac{1}{4} = \frac{36 - b^2}{36}\]
\[ \Rightarrow 36 = 144 - 4 b^2 \]
\[ \Rightarrow b^2 = 27\]
\[\text{ Substituting the values of } a^2 \text{ and } b^2 \text{ ,we get:} \]
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
\[ \Rightarrow \frac{x^2}{36} + \frac{y^2}{27} = 1\]
\[ \Rightarrow \frac{3 x^2 + 4 y^2}{108} = 1\]
\[ \Rightarrow 3 x^2 + 4 y^2 = 108\]
\[\text{ This is the required equation of the ellipse.} \]