Given:
Image of the ellipse
To find:
The equation of the image
Solution:
The given image is a ellipse.
Center of the ellipse = (0, 0)
x-axis points are (-3, 0) and (3, 0).
y-axis points are (2, 0) and (-2, 0).
Standard form of equation of ellipse:
[tex]$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1[/tex]
where (h, k) is the center = (0,0)
a is the point on x-axis where y = 0. Hence a = 3.
b is the point on y-axis where x = 0. Hence b = 2.
Substitute this in the standard form of ellipse.
[tex]$\frac{(x-0)^{2}}{3^{2}}+\frac{(y-0)^{2}}{2^{2}}=1[/tex]
[tex]$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1[/tex]
To make the denominator same multiply 1st term by [tex]\frac{4}{4}[/tex] and 2nd term by [tex]\frac{9}{9}[/tex].
[tex]$\frac{4x^{2}}{4\times9}+\frac{9y^{2}}{9\times4}=1[/tex]
[tex]$\frac{4x^{2}}{36}+\frac{9y^{2}}{36}=1[/tex]
[tex]$\frac{4x^{2}+9y^{2}}{36}=1[/tex]
Multiply by 36 on both sides
[tex]$\frac{4x^{2}+9y^{2}}{36}\times 36=1\times 36[/tex]
[tex]${4x^{2}+9y^{2}}={36}[/tex]
The equation of the image is [tex]${4x^{2}+9y^{2}}={36}[/tex].
