Question

How do you find the vertex, the directrix, the focus and eccentricity of $ \dfrac{9{{x}^{2}}}{25}+\dfrac{4{{y}^{2}}}{25}=1 $ ?

Answer 1

Hint: In this question, we need to find the vertex, the directrix, the focus and eccentricity of given equation of ellipse. For this, we will first convert the equation into the equation of the form $ \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1 $ and find the values of a, b. If b>a, vertices along major axis will be given by $ \left( 0,\pm b \right) $ and along minor axis will be $ \left( \pm a,0 \right) $ . Eccentricity will be equal to $ e=\sqrt{1-\dfrac{{{a}^{2}}}{{{b}^{2}}}} $ . Focus will be along y axis (if b>a) and given by $ \left( 0\pm be \right) $ . Equation of directrix will be $ y=\pm \dfrac{b}{e} $ .

Complete step by step answer:
Here we are given the equation of ellipse as $ \dfrac{9{{x}^{2}}}{25}+\dfrac{4{{y}^{2}}}{25}=1 $ .
We know the general equation of the ellipse is in the form $ \dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1 $ . So let us convert the given equation into the general form, we get $ \dfrac{{{x}^{2}}}{\dfrac{25}{9}}+\dfrac{{{y}^{2}}}{\dfrac{25}{4}}=1 $ .
We know that $ {{\left( 5 \right)}^{2}}=25,{{\left( 3 \right)}^{3}}=9,{{\left( 2 \right)}^{2}}=4 $ . So we can write it as $ \dfrac{{{x}^{2}}}{{{\left( \dfrac{5}{3} \right)}^{2}}}+\dfrac{{{y}^{2}}}{{{\left( \dfrac{5}{2} \right)}^{2}}}=1 $ .
Comparing it with the general equation we get $ a=\dfrac{5}{3}\text{ and b}=\dfrac{5}{2} $ .
Now as we can see b>a becomes $ \dfrac{5}{3}\text{ }>\text{ }\dfrac{5}{2} $ . Therefore, the major axis will be along the y axis and the minor axis will be the x-axis. Let us now try to find all the asked values.
Vertex:
An ellipse has vertices, two along the major axis and two along the minor axis. We know that the major axis is along the y-axis, so vertices along the major axis will be $ \left( 0,\pm b \right) $ and the minor axis is along the x-axis, so vertices along the minor axis will be $ \left( \pm a,0 \right) $.
Thus putting in the values of a, b we get, vertices along major axis are $ \left( 0,\pm \dfrac{5}{2} \right),\left( 0,\pm 2.5 \right) $ . Vertices along minor axis are $ \left( \pm \dfrac{5}{3},0 \right),\left( \pm 1.67,0 \right) $ .
Eccentricity:
When b is greater than a, eccentricity is given by the formula, $ e=\sqrt{1-\dfrac{{{a}^{2}}}{{{b}^{2}}}} $ .
Putting in the values of a, b we get,
\[\begin{align}
  & e=\sqrt{1-\dfrac{{{\left( \dfrac{5}{3} \right)}^{2}}}{{{\left( \dfrac{5}{2} \right)}^{2}}}} \\
 & \Rightarrow e=\sqrt{1-\dfrac{\dfrac{25}{9}}{\dfrac{25}{4}}} \\
 & \Rightarrow e=\sqrt{1-\dfrac{25}{9}\times \dfrac{4}{25}} \\
 & \Rightarrow e=\sqrt{1-\dfrac{4}{9}} \\
\end{align}\]
Taking LCM as 9 we get,
 $ \begin{align}
  & e=\sqrt{\dfrac{9-4}{4}} \\
 & \Rightarrow e=\sqrt{\dfrac{5}{9}} \\
\end{align} $
We know that $ \sqrt{9}=3 $ so we get, $ e=\dfrac{\sqrt{5}}{3} $ .
Therefore, eccentricity of the ellipse is $ \dfrac{\sqrt{5}}{3} $ .
Focus:
Focus is present along the major axis. In this case, we have it along the y-axis. So focus are given by $ \left( 0,\pm be \right) $ .
Putting in the values of b we get,
 $ \text{Focii}=\left( 0,\pm \dfrac{5}{2}\times \dfrac{\sqrt{5}}{3} \right)=\left( 0,\pm \dfrac{5\sqrt{5}}{6} \right) $ .
Simplifying we get $ \dfrac{5\sqrt{5}}{6} $ . So $ \text{Focii}=\left( 0,\pm 1.86 \right) $ .
Directrix:
We have two equation of directrix i.e. $ y=\pm \dfrac{b}{e} $ . Putting in the values of b and e we get,
\[\begin{align}
  & y=\pm \dfrac{\dfrac{5}{2}}{\dfrac{\sqrt{5}}{3}} \\
 & \Rightarrow y=\pm \dfrac{5}{2}\times \dfrac{3}{\sqrt{5}} \\
 & \Rightarrow y=\pm \dfrac{3\sqrt{5}}{2} \\
\end{align}\]
Simplifying $\dfrac{3\sqrt{5}}{2}$ we get 3.354
Therefore, $y=\pm 3.354$ is the equation of two directrix.

Note:
Students should keep in mind to use both $\pm $ signs to find the value as the ellipse is symmetrical and has four vertices, two focii, and two directrices. Focii and directrix are always found along the major axis. Take care of calculation while finding eccentricity. The eccentricity of the ellipse is always less than 1.