Consider a family of conics that all have a focus at (0,1) and a vertex at the origin. Working in Real plane?

Question

a)find equations of two ellipses that have these properties
b)find equations of two different hyperbolas that have these properties
c)explain why there is only one parabola that satisfies these properties and find its equation
d)sketch the conics you found in parts a-c on the same coordinate axes
e)how are the ellipses and hyperboals related to the parabola

Answer 1

Consider a family of conics that all have a focus at (0,1) and a vertex at the origin. Working in Real plane?

a) Find equations of two ellipses that have these properties.
There are an infinite number of such ellipses. We will have to make two additional assumptions.

i) Let a - c = 1 and let e = 1/2
An ellipse has two foci. The assumption a - c = 1 assumes that we have been given the focus closest to the vertex. We also assumed the eccentricity.

The vertex and focus are along a vertical line, so the major axis is vertical.

e = c/a = 1/2
2c = a

a - c = 2c - c = c = 1
a = 2c = 2

b² = a² - c² = 4 - 1 = 3

The center of the ellipse is (h,k) = (0,1 + c) = (0,2).

The equation of the ellipse is

(x - h)²/b² + (y - k)²/a² = 1
x²/3 + (y - 2)²/4 = 1
__________

ii) a + c = 1 and let e = 1/2
An ellipse has two foci. The assumption a + c = 1 assumes that we have been given the focus farthest from the vertex. We also assumed the eccentricity.

The vertex and focus are along a vertical line, so the major axis is vertical.

e = c/a = 1/2
2c = a

a + c = 2c + c = 3c = 1
c = 1/3
a = 2c = 2/3
a² = 4/9

b² = a² - c² = 4/9 - 1/9 = 1/3

The center of the ellipse is (h,k) = (0,1 - c) = (0,2/3).

The equation of the ellipse is

(x - h)²/b² + (y - k)²/a² = 1
x²/(1/3) + (y - 2)²/(4/9) = 1
__________

b) Use a similar approach for the hyperbolas.

c) There is only one such parabola because a parabola only has one focus and the eccentricity is always one.

You can do the rest.

Answer 2

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Answer 3

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