Question

How do you find the centre, vertices, foci and eccentricity of $\dfrac{{{x}^{2}}}{100}+\dfrac{{{y}^{2}}}{64}=1$?

Answer 1

This question is from the topic of ellipse which belongs to the chapter co-ordinate geometry. We will first know the general equation of ellipse. After that, we will understand how to find centre, vertices, foci and eccentricity of ellipse from the general equation of ellipse. After that, we will find the centre, vertices, foci and eccentricity from the equation of ellipse which is given as $\dfrac{{{x}^{2}}}{100}+\dfrac{{{y}^{2}}}{64}=1$.

Complete step by step solution:
Let us solve this question.
In this question, we have asked to find out the terms like centre, vertices, foci and eccentricity from the given equation of ellipse $\dfrac{{{x}^{2}}}{100}+\dfrac{{{y}^{2}}}{64}=1$.
Let us first understand from the general equation of ellipse.
The general equation of ellipse is
$\dfrac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}+\dfrac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1$
From the above equation of ellipse, we can say that
The centre of the ellipse is (h,k).
Foci of ellipse will be (h+c,k) and (h-c,k)
Vertices of ellipse will be (h+a,k) and (h-a,k)
Eccentricity of ellipse will be $e=\dfrac{c}{a}=\dfrac{\sqrt{{{a}^{2}}-{{b}^{2}}}}{a}$
We can take reference from the following figure:
So, we have given equation of ellipse as $\dfrac{{{x}^{2}}}{100}+\dfrac{{{y}^{2}}}{64}=1$
We can write this equation as
$\dfrac{{{\left( x-0 \right)}^{2}}}{{{10}^{2}}}+\dfrac{{{\left( y-0 \right)}^{2}}}{{{8}^{2}}}=1$
Here, we can say that a=10, b=8, h=0, k=0
Let us first find out the value of c.
$c=\sqrt{{{a}^{2}}-{{b}^{2}}}=\sqrt{{{10}^{2}}-{{8}^{2}}}=\sqrt{100-64}=\sqrt{36}=6$
Now, we can say that
The centre of ellipse is (0,0)
Foci of ellipse are (6,0) and (-6,0)
Vertices of ellipse are (10,0) and (-10,0)
Eccentricity of ellipse is $e=\dfrac{6}{10}=0.6$
Now, we have found all the values that we had asked for.

Note: We should have a better knowledge in the topic of coordinate geometry. Remember the following formulas:
If general equation of ellipse is $\dfrac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}+\dfrac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1$.
If ${{a}^{2}}>{{b}^{2}}$, then
Foci of ellipse will be (h+c,k) and (h-c,k)
Vertex of ellipse will be (h+a,k) and (h-a,k)
Eccentricity of ellipse will be $e=\dfrac{c}{a}=\dfrac{\sqrt{{{a}^{2}}-{{b}^{2}}}}{a}$
And if ${{a}^{2}}<{{b}^{2}}$, then
Foci of ellipse will be (h,k+c) and (h,k-c)
Vertex of ellipse will be (h,k+b) and (h,k-b)
Eccentricity of ellipse will be $e=\dfrac{c}{b}=\dfrac{\sqrt{{{b}^{2}}-{{a}^{2}}}}{b}$
Where $c=\sqrt{\left| {{a}^{2}}-{{b}^{2}} \right|}$