a parabola has a vertex v=(6,4) and a focus f=(6,1) what is the equation
from first principles, the directrix is y = 7
let P(x,y) be any point
then PF = PD, where D is the vertical distance from P to the directrix
√( (x-6)^2 + (y-1)^2 ) = √(x-x)^2 + (y-7)^2 )
square and expand
x^2 - 12x + 4 + y^2 - 2y + 1 = y^2 - 14y + 49
x^2 - 12x - 44 = -14y
verification:
http://www.wolframalpha.com/input/?i=x%5E2+-+12x+-+44+%3D+-14y
Or, knowing that the equation of a parabola with focus at y=(0,0) and focus at y = -p is
x^2 = 4py
we have a shifted parabola with p=-3, so the equation is
(x-6)^2 = 4(-3)(y-4)
x^2-12x+36 = -12y+48
x^2-12x+84 = -12y
Verify at
http://www.wolframalpha.com/input/?i=parabola+%28x-6%29^2+%3D+4%28-3%29%28y-4%29
Oops, Reiny - if you add the word parabola to your function at wolframalpha, you will see where the focus and vertex lie. Must have had a typo somewhere in the math.
To identify the equation of a parabola given its vertex and focus, you can use the standard form equation for a vertical parabola, which is:
(x - h)^2 = 4p(y - k)
Where (h, k) represents the vertex coordinates, and p represents the distance between the focus and the vertex. In this case, the vertex is V = (6, 4), and the focus is F = (6, 1).
Step 1: Find the value of p
The distance between the focus and the vertex is the value of p. In this case, the y-coordinate of the vertex is 4, and the y-coordinate of the focus is 1, so p = 4 - 1 = 3.
Step 2: Substitute the values into the equation
Using the values of h, k, and p, we can substitute them into the equation in order to obtain the final equation:
(x - 6)^2 = 4 * 3 (y - 4)
Simplifying:
(x - 6)^2 = 12 (y - 4)
And that is the equation of the parabola whose vertex is V (6, 4) and focus is F (6, 1).