Answer:
a. a = 3, b = 2
b. The coordinate of the center is (2, -1)
c. The eccentricity of the ellipse is √5/3
d. Please see attached graph of the ellipse created with MS Excel
Step-by-step explanation:
7. a. The given equation of the ellipse is presented as follows;
4·x² + 9·y² - 16·x + 18·y - 11
The general equation of an ellipse
[tex]\dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1[/tex]
We can write;
4·x² - 16·x + 9·y² + 18·y - 11
4·(x² - 4·x + 4) + 9·(y² + 2·y + 1) - 25 - 11
4·(x² - 4·x + 4) + 9·(y² + 2·y + 1) = 25 + 11 = 36
4·(x - 2)² + 9·(y + 1)² = 36
[tex]\dfrac{4\cdot (x - 2)^2}{36} + \dfrac{9 \cdot (y + 1)^2}{36} = \dfrac{36}{36}[/tex]
[tex]\therefore \dfrac{ (x - 2)^2}{9} + \dfrac{ (y + 1)^2}{4} = \dfrac{ (x - 2)^2}{3^2} + \dfrac{ (y + 1)^2}{2^2} = 1[/tex]
By comparison, a = 3, b = 2, h = 2, k = -1
b. The coordinate of the center, (h, k) = (2, -1)
c. The eccentricity of the ellipse = c/a
c² = a² - b²
∴ c² = 3² - 2² = 5
c = √5
Eccentricity = √5/3
d. Please find attached the graph of the ellipse created with MS Excel
