my equation for an ellipse is x^2/1^2 + y^2/1.5^2 = 1. double the distance from the foci to the vertex. what is the new equation?
The equation of the ellipse is x^2/1^2 + y^2/1.5^2 = 1.
To find the distance from the foci to the vertex, we can use the formula c^2 = a^2 - b^2, where c is the distance from the center to the foci, and a and b are the lengths of the semi-major and semi-minor axes respectively.
In this case, a = 1.5 and b = 1. To find c, we rearrange the formula:
c^2 = a^2 - b^2
c^2 = 1.5^2 - 1^2
c^2 = 2.25 - 1
c^2 = 1.25
c = √1.25
c ≈ 1.12
The distance from the foci to the vertex is 2c, so:
2c = 2 * 1.12 = 2.24
To find the new equation, we double the distance from the foci to the vertex, which means the distance from the foci to the vertex becomes 2.24. We can substitute this value back into the formula c^2 = a^2 - b^2 to find the new value of b:
c^2 = a^2 - b^2
(2.24)^2 = 1.5^2 - b^2
5.02 ≈ 2.25 - b^2
b^2 ≈ 2.25 - 5.02
b^2 ≈ -2.77
Since b is squared, it cannot be negative. Therefore, there is no real solution for b^2 ≈ -2.77.
Thus, there is no real equation for the new ellipse with twice the distance from the foci to the vertex.