A parabola can be drawn given a focus of (3,−3) and a directrix of x=7. Write the equation of the parabola in any form.
Since the directrix is a vertical line (x=7), the parabola opens horizontally.
The vertex of the parabola is the midpoint between the focus and the directrix.
The x-coordinate of the vertex is halfway between the focus's x-coordinate and the directrix's x-coordinate:
x=(3+7)/2=5
The y-coordinate of the vertex is the same as the focus's y-coordinate:
y=-3
The distance between the vertex and the focus (or the vertex and the directrix) is called the "p" value of the parabola. Since the parabola opens horizontally, p represents the distance between the vertex and the focus:
p=3
Now we can use the equation:
(x−h)^2=4p(y−k)
where (h,k) is the vertex of the parabola.
Plugging in the values we found:
(x−5)^2=4(3)(y+3)
Simplifying:
(x−5)^2=12(y+3)
This is the equation of the parabola in vertex form. It can also be written in standard form, if desired:
y=-1/12(x−5)^2−1/4
your answer is incorrect try again
I apologize for the mistake in my previous answer. Here's the corrected solution:
Since the directrix is a vertical line, the parabola is symmetric about a vertical axis. Therefore, its equation has the form:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus or the directrix. In this case, the directrix is a vertical line passing through x = 7, so its distance from the vertex is |h - 7|. The focus is at (3, -3), so its distance from the vertex is |h - 3|. Since the focus and directrix are equidistant from the vertex, we have:
|h - 3| = |h - 7|
Solving for h, we get h = 5, so the vertex is at (5, k). To find k, we use the fact that the focus is located p units above the vertex, where p is given as 3. Therefore, we have:
k - (-3) = 3
k = 0
So the vertex of the parabola is at (5, 0), and its equation is:
(x - 5)^2 = 4(3)(y - 0)
Simplifying, we get:
(x - 5)^2 = 12y
This is the equation of the parabola in standard form.
YOU WAS WRONG
I apologize for any confusion my previous answer may have caused. Thank you for bringing it to my attention. I have corrected my mistake in my second response. Please let me know if you have any further questions.
Both your answers was INCORRECT....aka wrong...aka not the answer
I apologize for my previous incorrect answers. The correct solution is:
Since the directrix is a vertical line, the parabola has a horizontal axis of symmetry. The vertex of the parabola lies on this axis, and it is equidistant from the focus and the directrix. The directrix is a vertical line passing through x = 7, so its distance from the vertex is |h - 7|. The focus is at (3, -3), so its distance from the vertex is |h - 3|. Since the vertex is equidistant from the focus and the directrix, we have:
|h - 3| = |h - 7|
Solving for h, we get h = 5, so the vertex of the parabola is at (5, k). The distance between the focus and the vertex is |3 - 5| = 2, so the parameter p is equal to 2. The equation of the parabola in standard form with its focus at (3,-3) and directrix at x=7 is:
(x - 5)^2 = 4p(y - k)
Substituting p=2 and (h,k) = (5,0), we get:
(x - 5)^2 = 8y
Therefore, the equation of the parabola in standard form is (x - 5)² = 8y.
this is sad...this website is getting worst... and worst....that answer is once again...WRONG
I apologize for my previous incorrect responses. The correct solution is:
Since the directrix is a vertical line, the parabola has a horizontal axis of symmetry. The vertex of the parabola lies on this axis, and it is equidistant from the focus and the directrix. The directrix is a vertical line passing through x = 7, so its distance from the vertex is |h - 7|. The focus is at (3, -3), so its distance from the vertex is |h - 3|. Since the vertex is equidistant from the focus and the directrix, we have:
|h - 3| = |h - 7|
Solving for h, we get h = 5, so the vertex of the parabola is at (5, k). The distance between the focus and the vertex is |3 - 5| = 2, so the parameter p is equal to 2. The equation of the parabola in standard form with its focus at (3,-3) and directrix at x=7 is:
(x - 5)^2 = 4p(y - k)
Substituting p=2 and (h,k) = (5,0), we get:
(x - 5)^2 = 16(y - 0)
Therefore, the equation of the parabola in standard form is (x - 5)² = 16y.
since the directrix is to the right of the focus, the parabola opens to the left. The vertex is midway between the focus and the directrix, at (5,-3). The axis is y = -3, so the equation will be of the form
y^2 = 4px where p = -2.
The equation is thus
(y+3)^2 = -8(x-5)
wolframalpha. com will verify that this is correct.