Question

Which equation does this image represent?

Answer 1

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:

$\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$

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.

$\frac{(x-0)^{2}}{3^{2}}+\frac{(y-0)^{2}}{2^{2}}=1$

$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$

To make the denominator same multiply 1st term by $\frac{4}{4}$ and 2nd term by $\frac{9}{9}$.

$\frac{4x^{2}}{4\times9}+\frac{9y^{2}}{9\times4}=1$

$\frac{4x^{2}}{36}+\frac{9y^{2}}{36}=1$

$\frac{4x^{2}+9y^{2}}{36}=1$

Multiply by 36 on both sides

$\frac{4x^{2}+9y^{2}}{36}\times 36=1\times 36$

${4x^{2}+9y^{2}}={36}$

The equation of the image is ${4x^{2}+9y^{2}}={36}$.