A parabola can be drawn given a focus of (9, -2)(9,−2) and a directrix of x=3x=3. Write the equation of the parabola in any form.
I will guess you meant:
A parabola can be drawn given a focus of (9, -2) and a directrix of x=3. Write the equation of the parabola in any form.
Using the actual definition:
let P(x,y) be any point on the parabola
Distance of P from focal point = distance of P to the directrix
√((x-9)^2 + (y+2)^2) = x - 3
square both sides and simplify
x^2 - 18x + 81 + y^2 + 4y + 4 = x^2 - 6x + 9
y^2 + 4y + 76 = 12x
Recall that the parabola
y^2 = 4px has
focus at (p,0)
directrix x = -p
The vertex is midway between the directrix and the focus.
So your parabola has a vertex at (6,-2) and p=3
That makes its equation
(y+2)^2 = 12(x-6)
confirmation is at
https://www.wolframalpha.com/input/?i=parabola+%28y%2B2%29%5E2+%3D+12%28x-6%29
A parabola can be drawn given a focus of (2,6) and a directrix of x=8. What can be said about the parabola?
To write the equation of a parabola given its focus and directrix, we first need to determine the vertex of the parabola. The vertex lies at the midpoint between the focus and the directrix.
Given that the focus is (9, -2) and the directrix is x = 3, we can find the x-coordinate of the vertex by averaging the x-coordinates of the focus and the directrix. In this case, (9 + 3) / 2 = 12 / 2 = 6. The y-coordinate of the vertex will be the same as that of the focus, which is -2. Therefore, the vertex is (6, -2).
Since the directrix is a vertical line, the parabola will open either upward or downward. In this case, since the directrix is below the focus, the parabola will open downward.
Now that we have the vertex, we can write the equation of the parabola in vertex form, which is (x - h)^2 = 4p(y - k), where (h, k) represents the vertex and p represents the distance between the vertex and either the focus or the directrix.
In our case, the vertex is (6, -2). To find the value of p, we can use the distance formula between the vertex and the focus. The formula is p = |x - h|, where h is the x-coordinate of the vertex. In this case, p = |9 - 6| = |3| = 3.
Therefore, the equation of the parabola in vertex form is (x - 6)^2 = 4(-3)(y - (-2)). Simplifying the equation, we have (x - 6)^2 = -12(y + 2). That is the equation of the parabola given a focus of (9, -2) and a directrix of x = 3.