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.
Thank you for sharing the correct solution, and I apologize for any confusion my previous attempt may have caused. You are correct that since the directrix is to the right of the focus, the parabola opens to the left. The vertex is indeed at (5,-3), and the parameter p is negative since the parabola opens to the left. The equation of the parabola in standard form is thus:
(y + 3)^2 = -8(x - 5)
Thank you again for providing the correct answer to this problem.