The directrix of a parabola is y=9 . The focus of the parabola is (2,5) .
What is the equation of the parabola?
y=−1/8(x−2)^2+7
y=1/8(x−2)^2−7
y=1/8(x−2)^2+7
y=−1/8(x−2)^2−7
They are different questions.
I don't understand it well.
The parabola x^2 = 4py has
focus = (0,p)
directrix y = -p
The distance between focus and directrix is 2p
For your parabola, that distance is 5-9 = -4 = 2p, so the usual equation would be
x^2 = -8y
But the focus is at (2,5) and not (0,-2). So it has been shifted right 2 and up 7. That makes the equation
(x-2)^2 = -8(y-7)
So pick the choice that matches.
To find the equation of a parabola given its directrix and focus, we can use the formula:
(y - yf)^2 = 4p(x - xf)
where (xf, yf) is the coordinates of the focus and p is the distance from the vertex to either the focus or the directrix.
In this case, the coordinates of the focus are (2, 5), and the equation of the directrix is y = 9. The distance between the vertex and the focus is the same as the distance between the vertex and the directrix. So, we can find the value of p by calculating the distance between the vertex and the directrix.
The distance between a point (x, y) and a line Ax + By + C = 0 can be found using the formula:
distance = |Ax + By + C| / sqrt(A^2 + B^2)
For the given directrix equation y = 9, A = 0, B = 1, and C = -9. Substituting these values into the formula, we have:
distance = |0x + 1y - 9| / sqrt(0^2 + 1^2)
distance = |y - 9|
Since the value of p is the same as the distance between the vertex and the directrix, we have:
p = |y - 9|
Now, we can substitute the values of p and (xf, yf) into the equation of the parabola:
(y - yf)^2 = 4p(x - xf)
Substituting (2, 5) for (xf, yf) and p = |y - 9|, we get:
(y - 5)^2 = 4|y - 9| * (x - 2)
Now, we simplify and isolate y to obtain the equation of the parabola.
geeesssh, I know that
The method is the same!
same as
https://www.jiskha.com/questions/1815358/the-directrix-of-a-parabola-is-y-4-the-focus-of-the-parabola-is-2-2-what-is
1. Look for examples in your book
2. Follow method used in class.
3. Explain to your teacher what part you don't understand.