Question

How do you find the equation of an ellipse with vertices $ \left( {0, \pm 8} \right) $ and the foci $ \left( {0, \pm 4} \right) $ ?

Answer 1

Hint: In the given question, we are required to find the equation of the ellipse. We are provided the foci and vertices of the ellipse whose equation has to be found. We first determine the major and minor axis of the ellipse and then find the length of major and minor axis accordingly with the use of information given to us.

Complete step by step solution:
So, we have an ellipse whose foci are $ \left( {0,4} \right) $ and $ \left( {0, - 4} \right) $ and vertices are $ \left( {0,8} \right) $ and $ \left( {0, - 8} \right) $ . We have to find the equation of the ellipse.
We notice that both of the foci lie on y axis as well as both vertices also lie on y axis.
This means that the major axis is the y axis.
Now, the center of the ellipse is the midpoint of the vertices of the ellipse.
Hence, we can find the center of the ellipse using the midpoint formula.
Using the midpoint formula, midpoint of two points can be calculated as $ \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) $ .
So, the center of the ellipse is $ \left( {\dfrac{{0 + 0}}{2},\dfrac{{ - 8 + 8}}{2}} \right) = \left( {0,0} \right) $ .
Hence, the equation of the ellipse is of the form $ \dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1 $ .
Now, we have to find the values of a and b in order to find the complete equation of the ellipse.
Now, we know that foci of an ellipse with its major axis along y axis are $ \left( {0, \pm ae} \right) $ , where e is the eccentricity of the ellipse.
So, we get, $ ae = 4 - - - - - \left( 1 \right) $ .
Also, the length of major axis $ = 2a $
So, the distance between vertices $ \left( {0, \pm 8} \right) $ of the ellipse is $ 16 $ units.
Hence, we get, $ 2a = 16 $
 $ \Rightarrow a = 8 $
Now, we can find the value of e by substituting the value of a in equation $ \left( 1 \right) $ .
So, we get, $ \left( 8 \right)e = 4 $
 $ \Rightarrow \left( 8 \right)e = 4 $
 $ \Rightarrow e = \dfrac{4}{8} = \dfrac{1}{2} $
Now, we know that $ {a^2}\left( {1 - {e^2}} \right) = {b^2} $
So, we get,
 $ \Rightarrow {8^2}\left( {1 - {{\left( {\dfrac{1}{2}} \right)}^2}} \right) = {b^2} $
Simplifying further, we get,
 $ \Rightarrow {8^2}\left( {1 - \dfrac{1}{4}} \right) = {b^2} $
 $ \Rightarrow {8^2}\left( {\dfrac{3}{4}} \right) = {b^2} $
 $ \Rightarrow {b^2} = 8 \times 8 \times \dfrac{3}{4} $
 $ \Rightarrow {b^2} = 48 $
Taking square on root both sides of the equation, we get,
 $ \Rightarrow b = 4\sqrt 3 $
So, the complete equation of the ellipse is $ \dfrac{{{x^2}}}{{64}} + \dfrac{{{y^2}}}{{48}} = 1 $

Note: There are various ways of finding the equation of an ellipse depending on what terms and conditions are given in the problem. The problem requires the basic knowledge of coordinate geometry. Applications of algebraic rules such as transposition and BODMAS are essential for solving such questions.