State the focus and directrix

Question

State the focus and directrix For

x = y^2

Answer 1

when the y^2 term is positive, the parabola opens to the right, so the focus has the same y coordinate as the vertex.

The vertex is the origin if there are no constants in the equation, (0,0)

the focus is (x,0) Problem is to find the x coordinate.

the general formula is x=4ay^2 where a is the distance from the vertex to the focus, or in this case, the x coordinate. x=y^2 so 4a = 1 and a = 1/4

the focus is (1/4,0)

the directrix is the same distance on the other side of the origin or x=-1/4

the parabola is the collection of all points equidistant from the focus and the directrix line

take any point on the parabola such as (4,2) 4=2 squared.

the distance to the directrix to that point is 4 1/4

the distance to the focus is the distance from (4,2) to (1/4,0)

that distance squared is the sum of squares of the differences in x and y coordinates of the two points:

2 squared is 4, 4-1/4 = 3 3/4 = 15/4 15/4 squared = 225/16

4=64/16

225/16 +64/16 = 289/16

the square root of 289/16 = 17/4 = 4 1/4

Answer 2

(y - k)² - 4p(y - h)

Since we have no value for h or k, they each equal 0, so our vertex is (0,0).

Remember that a variable with no coefficient has an implied 1, but just like we don't call you 1 Alyssa, we don't say 1x.

4p = 1

p = 1/4

Since p > 0, our parabola opens to the right, and our directrix is x = -1/4.

The focus is (p,0), so (1/4,0).