Write an equation of a parabola with vertex (2,0) and directrix x=4 . Write your final answer in standard conic form.
You will be graded on finding the correct equation in the correct form and showing correct, step-by-step work.
Given that the vertex is (2,0) and the directrix is x=4, the focus is located at the same distance from the vertex but in the opposite direction. Since the directrix is a vertical line, the focus will lie on a horizontal line passing through the vertex.
The distance between the vertex and the directrix is 2 units (from x=2 to x=4), so the focus will be located 2 units to the left of the vertex. Therefore, the focus is at (0,0).
The standard form of a parabola with the focus at (h+k, k) is:
(y-k)^2 = 4p(x-h)
Where p is the distance between the vertex and focus. In this case, p = 2.
Plugging in the values of the vertex and focus, we get:
(y-0)^2 = 4(2)(x-2)
y^2 = 8(x-2)
y^2 = 8x - 16
Therefore, the equation of the parabola in standard conic form is:
y^2 = 8x - 16