A parabola can be drawn given a focus of left bracket, 5, comma, 6, right bracket(5,6) and a directrix of y, equals, minus, 10y=−10. What can be said about the parabola?

Answer 1

Since the focus is (5,6) and the directrix is y=-10, it represents a vertical parabola opening upward and:

p = (6 - (-10)) / 2 = 8 (distance from focus to point on parabola or directrix to point on parabola)

(h.k) is the vertex point

directrix = k - p

-10 = k - 8

k = -2

h=5 (same x value as the focus)

vertex is (h,k) = (5,-2)

genral equation and vertex form of the equations:

(x - h)^2 = p(y - k)^2, or y = a(x - h)^2 + k

(x - 5)^2 = 8(y +2)^2, or y=(1/32)(x-5)^2 - 2

Answer 2

The axis of symmetry is perpendicular to the directrix and the focus lies on the axis of symmetry. So, the axis of symmetry is the vertical line x = 5. The vertex of the parabola lies on the axis of symmetry and is halfway between the focus and directrix. So, the vertex is (5, -2). The focus lies above the vertex, so the parabola opens upward.

Equation of parabola has the form y = a(x - 5)² - 2.

a > 0 and a = 1/(4p), where p is the distance from the vertex to the focus. So, p = 8 and a = 1/32.

Note: p is positive since the focus lies above the vertex. If the focus had been below the vertex, then p would have been negative.

Equation of parabola: y = (1/32)(x - 5)² - 2

Another approach is to use the definition of parabola: A parabola is the set of all points equidistant from the focus and directrix.

If we let (x,y) be a point on the parabola, than using the distance formula and the definition of parabola and squaring both sides, we have:

(x-5)² + (y-6)² = (x-x)² + (y-(-10))²

So, (x-5)² + y² - 12y + 36 = y² + 20y + 100

(x-5)² -32y - 64 = 0

-32y = -(x-5)² + 64

So, y = (1/32)(x-5)² - 2