The eccentricity of the ellipse 9x2+ 5y2– 30y = 0 is:a)1/3b)2/3c...
9x2 + 5y2 - 30y = 0
9x2 + 5(y−3)2 = 45
We can write it as : [(x-0)2]/5 + [(y-3)2]/9 = 1
Compare it with x2/a2 + y2/b2 = 1
e = [(b2 - a2)/b2]½
e = [(9-5)/9]1/2
e = (4/9)½
e = ⅔
The eccentricity of the ellipse 9x2+ 5y2– 30y = 0 is:a)1/3b)2/3c...
To find the eccentricity of the ellipse, we need to first put the given equation in the standard form of an ellipse, which is:
(x^2/a^2) + (y^2/b^2) = 1
where 'a' and 'b' are the semi-major and semi-minor axes of the ellipse, respectively.
1. Rearrange the equation:
9x^2 + 5y^2 + 30y = 0
2. Complete the square for the y terms:
9x^2 + 5(y^2 + 6y) = 0
3. Divide the coefficient of x^2 by the constant term to determine 'a':
a^2 = -30/9
a^2 = -10/3
a = sqrt(-10/3) (imaginary)
4. Divide the coefficient of y^2 by the constant term to determine 'b':
b^2 = -5/9
b = sqrt(-5/9) (imaginary)
Since 'a' and 'b' are imaginary, this means that the given equation does not represent a real ellipse, but rather a degenerate ellipse or a pair of intersecting lines.
5. Calculate the eccentricity (e):
The eccentricity of an ellipse is defined as the square root of (1 - (b^2/a^2)).
e = sqrt(1 - (b^2/a^2))
e = sqrt(1 - (-5/9) / (-10/3))
e = sqrt(1 + 5/6)
e = sqrt(11/6)
Therefore, the eccentricity of the given ellipse is sqrt(11/6), which is approximately 0.8819. None of the given options (a, b, c, d) match this value, so the correct answer is None of these.