Question

How do find the center and vertices of the ellipse \[4{x^2} + 9{y^2} = 36\]?

Answer 1

Hint: We use the given equation and divide the complete equation by constant term on the right hand side of the equation such that the right hand side becomes 1. Now we write the denominators of the fractions formed on the left hand side of the equation in the form of a square of some numbers. Now we compare the equation formed with the general equation of the ellipse and find the vertices and center.
* An ellipse has two axes of symmetry, longer axis called major axis and shorter axis called minor axis. Each endpoint of the major axis is the vertex of the ellipse and similarly, each endpoint of minor axis is co-vertex. Center of the ellipse is the midpoint of the major and minor axis.
General equation of ellipse centered at origin and having major axis parallel to x-axis is \[\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1\], where coordinates of vertices are \[( \pm a,0)\].

Complete step-by-step solution:
We are given an equation \[4{x^2} + 9{y^2} = 36\] … (1)
Since we have to find the vertices and center of the ellipse, we will make this equation similar to the general form of the ellipse.
We have to make the right hand side of the equation as 1, so we will divide both sides of the equation by 36.
Divide both sides of equation (1) by 36
\[ \Rightarrow \dfrac{{4{x^2} + 9{y^2}}}{{36}} = \dfrac{{36}}{{36}}\]
Cancel common factors from numerator and denominator on both sides of the equation
\[ \Rightarrow \dfrac{{{x^2}}}{9} + \dfrac{{{y^2}}}{4} = 1\]
Now we know that \[4 = {2^2}\]and \[9 = {3^2}\]
\[ \Rightarrow \dfrac{{{x^2}}}{{{3^2}}} + \dfrac{{{y^2}}}{{{2^2}}} = 1\] … (2)
Now we compare equation (2) with general equation of the ellipse centered at origin and having major axis parallel to x-axis i.e. \[\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1\].
Then \[a = 3;b = 2\]
So, the vertices of the ellipse are\[( \pm 3,0)\]i.e. \[( - 3,0)\]and \[(3,0)\].

\[\therefore \]Center of the ellipse\[4{x^2} + 9{y^2} = 36\] is \[(0,0)\]and vertices of the ellipse \[4{x^2} + 9{y^2} = 36\] are \[( - 3,0)\]and \[(3,0)\]

Note:
Many students make the mistake of writing the value of vertices as \[(a,0)(0,b)\]as they think vertices means more than one vertex. Keep in mind the value of vertex on the major axis is called vertex, whereas the one on minor axis is called co-vertex.