Question

Find the eccentricity of the ellipse 64x^2 + 4y^2 = 256.

A) 1/4
B) 1/2
C) 1
D) 2

Answer 1

The eccentricity of the ellipse 64x^2 + 4y^2 = 256 is i√15.

Step-by-step explanation:

To find the eccentricity of the ellipse, we need to rewrite the equation of the ellipse in standard form. The standard form of an ellipse equation is:

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1,

where (h, k) is the center of the ellipse, and a and b are the semi-major and semi-minor axes respectively. Comparing this with the given equation 64x^2 + 4y^2 = 256, we can see that the equation is already in standard form, and the values of a and b are easily obtained as:

a = √(256/64) = 2

b = √(256/4) = 8

The eccentricity, represented by (e), can be calculated using the formula e = √(a^2 - b^2) / a. Substituting the values of a and b into this formula, we get:

e = √(2^2 - 8^2) / 2 = √(4 - 64) / 2 = √(-60) / 2 = √(-1) * √(60) / 2 = i√60 / 2 = √60 / 2i = (√60/2) * (1/i)

Therefore, the eccentricity of the ellipse is i√15.