What is the equation?

Question

Fcous is (5,7) and the directix

y=0 what is the the equation

Answer 1

focus (5,7) directrix y=0

it's an upward opening parabola with line of symmetry x=5

the vertex is midway between the focus and directrix V(5, 3.5)

just one more point and you have the quadratic equation

y = a(x-5)^2 + 7/2.

plug in that point and solve for "a" coefficient of the x^2 term

directly, horizontally, to the right (and left) of the focus is another point on the parabola

to the right is a point with y coordinate = 7, and x coordinate such that x=7+5=12

7 = a(7-5)^2 +3.5

7/2 = 49a

7a = 1/2

a = 1/14

y = (1/14)(x-5)^2 + 7/2 in vertex form

or 1/4p = a where p = 1/2 the distance from focus to directrix

= 1/4(3.5) = 1/14 = a

y =(1/4p)(x-5)^2 +7/2

y = (1/14)(x-5)^2 + 7/2

Answer 2

A parabola that has a directrix in the form of y = ____ (example, y = 0), will be in the form of:

4p(y - k) = (x - h)² and if it opens "up" then the "p" value is positive or if it opens "down" then the "p" value is negative.

Draw a sketch of what you were given.

On your sketch, you will notice that the focus is above the directrix. The parabola will always have its vertex between the focus and directrix and will open in the direction towards the focus. Your sketch will show that the parabola must open upwards, meaning the parabola fits the criteria I described above where the equation fits the form:

4p(y - k) = (x - h)²

and the "p" value is positive.

The directrix is a line and the focus is a point. Draw a line perpendicular to the directrix that goes through the focus. The vertex is on that line and it is halfway between the two. the vertex is the point (h, k) so now you know the value of both "h" and "k"

Once you get the vertex, you can calculate the "p" value. The "p" value is the distance between either the focus and the vertex, or the vertex and the directrix (along that perpendicular line).

Now, plug the values you got for "h", "k", and "p" into the generic equation for a parabola:

4p(y - k) = (x - h)²