Question

How do you find the standard form of the equation of the ellipse given the properties foci $\left( {0, \pm 5} \right)$, vertices$\left( {0, \pm 8} \right)$?

Answer 1

Hint: Given the values of vertices and foci. We have to determine the standard form of the equation of an ellipse. First, we will identify the coordinates of vertices and foci. If the vertices are $\left( {0, \pm a} \right)$ and foci are $\left( {0, \pm c} \right)$, then apply the equation of an ellipse in which the major axis is parallel to the y-axis. Then calculate the value of b using the values of vertices and foci. Then substitute the values into the standard equation of an ellipse.

Formula used:
The standard form of the equation of an ellipse with centre origin is given by:
$\dfrac{{{x^2}}}{{{b^2}}} + \dfrac{{{y^2}}}{{{a^2}}} = 1$
If the coordinates of the vertices are $\left( {0, \pm a} \right)$and coordinates of the foci are $\left( {0, \pm c} \right)$
The formula to find the coordinates of b using foci and vertices is given by:
\[ \Rightarrow {c^2} = {a^2} - {b^2}\]

Complete step-by-step answer:
We are given the vertices and foci $\left( {0, \pm 8} \right)$ and $\left( {0, \pm 5} \right)$ respectively. Here, the foci are on the y-axis which means the major axis is parallel to y-axis.
Now, determine the value of $a$ using the coordinates of the vertices.
\[ \Rightarrow a = \pm 8\]
Compute the value of \[{a^2}\] by squaring both sides.
\[ \Rightarrow {a^2} = {\left( { \pm 8} \right)^2}\]
\[ \Rightarrow {a^2} = 64\]
Now, determine the value of $c$ using the coordinates of the foci.
\[ \Rightarrow c = \pm 5\]
Compute the value of \[{c^2}\] by squaring both sides.
\[ \Rightarrow {c^2} = {\left( { \pm 5} \right)^2}\]
\[ \Rightarrow {c^2} = 25\]
Now, compute the value of \[{b^2}\] by substituting the values of \[{c^2}\] and \[{a^2}\] into the equation \[{c^2} = {a^2} - {b^2}\]
\[ \Rightarrow 25 = 64 - {b^2}\]
Now, add \[{b^2}\] on both sides of the equation.
\[ \Rightarrow 25 + {b^2} = 64 - {b^2} + {b^2}\]
\[ \Rightarrow 25 + {b^2} = 64\]
Now, subtract \[25\]from both sides of the equation.
\[ \Rightarrow 25 + {b^2} - 25 = 64 - 25\]
\[ \Rightarrow {b^2} = 39\]
Now, we will substitute the values into the standard form of an equation of ellipse.
$\dfrac{{{x^2}}}{{39}} + \dfrac{{{y^2}}}{{64}} = 1$

Hence, the standard form of the equation of the ellipse is $\dfrac{{{x^2}}}{{39}} + \dfrac{{{y^2}}}{{64}} = 1$.

Note:
In such types of questions the students mainly don't get an approach on how to solve it. In such types of questions students mainly forget to check whether the foci are on the x-axis or y-axis. Then, students can forget to find the center of the ellipse and apply the standard equation according to the center.