Question

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).

Answer 1

The general equation of an ellipse:

`((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) =1`

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


`((x+1)^2)/(9) + ((y-2)^2)/(16) =1`
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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 :

`((x-h)^2)/(a^2) - ((y-k)^2)/(b^2) =1`
the equation of the hyperbole will be

`((x-7)^2)/(4) - ((y-3)^2)/(12) =1`