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likoan [24]
3 years ago
6

The equation for the ellipse given is 4x^2+9y^2-16x+18y-11​

Mathematics
1 answer:
natita [175]3 years ago
8 0

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

\dfrac{(x - h)^2}{a^2}  + \dfrac{(y - k)^2}{b^2} = 1

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

\dfrac{4\cdot (x - 2)^2}{36}  + \dfrac{9 \cdot (y + 1)^2}{36} = \dfrac{36}{36}

\therefore \dfrac{ (x - 2)^2}{9}  + \dfrac{  (y + 1)^2}{4} =  \dfrac{ (x - 2)^2}{3^2}  + \dfrac{  (y + 1)^2}{2^2}  = 1

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

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