Final answer:
- The value of c is sqrt(5).
- The value of a is sqrt(216).
- The value of c/ a is sqrt(5)/3.
- The dilation factor is the ratio of their semi-major axes, which remains the same for similar figures. Therefore, the two ellipses have the same eccentricity.
- In this case, the converse is not true.
Step-by-step explanation:
To show that there is a dilation that transforms the equation 9x^2 + 4y^2 = 36 onto 9x^2 + 4y^2 = 225, we need to find the values of c and a for each ellipse.
The distance between the foci (c) can be found using the formula c^2 = a^2 - b^2, where a and b are the lengths of the semi-major and semi-minor axes respectively.
For the first ellipse, c^2 = 3^2 - 2^2 = 5, so c = sqrt(5).
Similarly, for the second ellipse, c^2 = 15^2 - 3^2 = 216, so c = sqrt(216).
The semi-major axis (a) is the distance from the center of the ellipse to the vertices.
For the first ellipse, a = 3, and for the second ellipse, a = 15.
The ratio c/a is the eccentricity of the ellipse. For both ellipses, the eccentricity is c/a = sqrt(5)/3.
The similarity of the ellipses can be explained by the fact that a dilation is a transformation that changes the size of an object, while preserving its shape.
In this case, the dilation factor is the ratio of their semi-major axes, which remains the same for similar figures.
Therefore, the two ellipses have the same eccentricity. However, the converse is not true. Two ellipses can have the same eccentricity but different shapes, depending on the value of the semi-minor axis (b).