Equations in standard form for conic figures! pleaseeee help!!!?

Question

i have no idea how to write an equation in standard form! im so lost! if someone could help me pleasee that would be amazing! you dont have to do the whole thing just as much as you can do :] thankyou so much!

write an equation in standard form for each of the following conic figures

1.an ellipse with a center at the origin, one vertex at (0,5) and one focus at (0,2)

2.a parabola with vertex (1,2) and directrix y = -2

3.a hyperbola with foci (0,-5) and (0,5) and difference of focal radii 8.

4.a circle with center (2,3) passing through (-3,5).

5. a parabola with focus (-1,0) and directrix x=1

Answer 1

I'll do a couple to start with:

1. The standard form of an ellipse of this type is:

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

Center: (h, k) = (0, 0)

a = distance from center to vertex = 5, so a^2 = 25

c = distance from center to focus = 2, so c^2 = 4

Relationship between a, b, c for ellipses: a^2 = b^2 + c^2

Solve for b^2:

25 = b^2 + 4

b^2 = 21

Equation:

(x - 0)^2/21 + (y - 0)^2/25 = 1

x^2/21 + y^2/25 = 1

4.The standard form of circle is:

(x - h)^2 + (y - k)^2 = r^2

Center = (h, k) = (2, 3)

Radius is the distance from the center to a point, so use the distance formula:

d^2 = r^2 = (x2 - x1)^2 + (y2 - y1)^2

(-3 - 2)^2 + (5 - 3)^2

(-5)^2 + (2)^2

25 + 4

29

(x - 2)^2 + (y - 3)^2 = 29

Hope that helps a bit.