1.Write the equation for an ellipse with vertices (–4, 2), (2, 2), (–1, –2) and (–1, 6).2. Write the equation for a hyperbola with vertices (9, 3) and (5, 3) and with foci (11, 3) and (3, 3).
Accepted Solution
A:
The general equation of an ellipse: [tex] \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} =1
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we have vertices (–4, 2), (2, 2), (–1, –2) and (–1, 6).⇒⇒⇒ red points by graphing the points ⇒⇒⇒ attached figure 1 the majority axis is the line connecting (–1, –2) and (–1, 6) and has a distance = 8 the minority axis is the line connecting (–4, 2), and (2, 2) and has a distance = 6 (h,k) represents the center of ellipse which is the intersection between axes ∴(h,k) = (-1,2) and a = 3 , b = 4 ∴ the equation of the ellipse is
A hyperbola with vertices (9, 3) and (5, 3) ⇒⇒⇒⇒ blue points and with foci (11, 3) and (3, 3).
⇒⇒⇒⇒ red points by graphing the points ⇒⇒⇒ attached figure 2 so, the hyperbole axis is horizontal (h,k) represents the center of hyperbola = (7,3) ⇒⇒⇒ green point a = distance between center and any of vertices = 7 - 5 = 2 c = distance between center and any of foci = 7 - 3 = 4 ∵ c² = a² + b² ∴ b² = c² - a² = 16 - 4 = 12 the general equation of the hyperbole : [tex] \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1
[/tex] the equation of the hyperbole will be [tex] \frac{(x-7)^2}{4} - \frac{(y-3)^2}{12} =1
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