(a) Sketch the conic section. Find and label any foci, vertices, and asymptotes. (x - 3)^2 – 9y^2 = 36
(b) Find the equation of the ellipse with foci (0,+2) and semi-major axis length 3.
Answers
a) the vertices are (9, 0) and (-3, 0).
the foci are (3 ± 2√10, 0)
the asymptotes are y = ±x/3 - 1
b) the equation of the ellipse is x² + (y-√5/2)² = 5/4
a) To find the foci, vertices, and asymptotes of the ellipse (x - 3)² - 9y² = 36, we can first divide both sides by 36 to get:
[tex]\frac{(x-3)^2}{36} - \frac{y^2}{4}=1[/tex]
Therefore, the center of the ellipse is (3, 0).
The semi-major axis length is √36 = 6, and the semi-minor axis length is √4 = 2.
Therefore, the vertices are (3 ± 6, 0) = (9, 0) and (-3, 0).
The foci are located at a distance of √(6²-2²) = 2√10 from the center along the major axis. Therefore, the foci are (3 ± 2√10, 0) and the equation of the major axis is x = 3.
To find the asymptotes, we will use the formula:
[tex]\frac{y-k}{b} = \pm\frac{x-h}{a}[/tex]
where (h, k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis. Therefore, the equations of the asymptotes are:
(y-0)/2 = ±(x-3)/6
y = ±x/3 - 1
b) To find the equation of the ellipse with foci (0, 2) and semi-major axis length 3, we can first find the center of the ellipse. Since the foci are located on the y-axis, the center must also be located on the y-axis. Therefore, the center is (0, c), where c is the distance between the center and one of the foci.
Since the semi-major axis length is 3, the distance between the center and one of the vertices is 3. Therefore, we have:
c² + (3/2)² = (3/2+2)²
c² = 5/4
Therefore, the center of the ellipse is (0, √5/2). The distance between the center and one of the foci is √5/2 - 2. Therefore, the distance between the center and one of the vertices is √{(√5/2)² - (√5/2 - 2)²} = √5.
Therefore, the semi-minor axis length is √5/2, and the equation of the ellipse is:
[tex]\frac{x^2}{\frac{5}{4} } +\frac{(y-\frac{\sqrt{5}}{2} )^2}{\frac{5}{4} } =1[/tex]
x² + (y-√5/2)² = 5/4
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Answer:
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