Find the equation of the parabola with directix y=3 and focus (5,7)?

Hi Anna.

Didn't have the time to actually answer the question that you wanted answering but i thought i'd direct you to a very good website that would give you a re-cap about Parabolas if that is ok. It is: Parabolas Lessons 1 to 4 by Jack Sarfaty.
You should find Jack Sarfaty's information about Parabolas very useful.

To find the equation of the parabola with a given directrix and focus, we can use the standard formula:

(x - h)^2 = 4p(y - k)

where (h, k) represents the coordinates of the vertex and p represents the distance between the vertex and the focus (or directrix).

Given that the directrix is y = 3 and the focus is (5, 7), we can determine the values of (h, k) and p.

Step 1: Identify the vertex (h, k)
The vertex lies halfway between the directrix and the focus. In this case, since the directrix is y = 3 and the focus is (5, 7), the vertex will have a y-coordinate of (3 + 7) / 2 = 5.

So, the vertex is (h, k) = (5, 5).

Step 2: Calculate the value of p
The value of p is the distance between the vertex and the focus (or directrix). As the vertex is (5, 5) and the focus is (5, 7), the distance is 2 units.

So, p = 2.

Step 3: Construct the equation
Using the values of (h, k) = (5, 5) and p = 2, we can form the equation of the parabola:

(x - 5)^2 = 4(2)(y - 5)

Simplifying, we have:

(x - 5)^2 = 8(y - 5)

Therefore, the equation of the parabola with directrix y = 3 and focus (5, 7) is (x - 5)^2 = 8(y - 5).

To find the equation of a parabola with given directrix and focus, you can use the definition of a parabola.

A parabola is defined as the set of all points that are equidistant from the focus and the directrix.

In this case, the given directrix is y = 3, and the focus is (5,7). We need to find the equation of the parabola.

Step 1: Determine the vertex of the parabola.
Since the directrix is a horizontal line and the focus is above the directrix, the vertex of the parabola will be halfway between the focus and the directrix, along the line perpendicular to the directrix.

The perpendicular line to the directrix y = 3 passing through the focus (5,7) intersects the directrix at the point (5,3) which is always equidistant from the focus. Therefore, the vertex is located at the midpoint between the focus and the point of intersection of the directrix and the perpendicular line.

The coordinates of the vertex is the average of the x-coordinates and the y-coordinates:
Vertex = ( (5 + 5)/2, (7 + 3)/2 )
= (5, 5)

Step 2: Determine the distance between the vertex and the focus.
Using the distance formula, the distance between the vertex (5,5) and the focus (5,7) is:
d = sqrt( (x2 - x1)^2 + (y2 - y1)^2 )
= sqrt( (5 - 5)^2 + (7 - 5)^2 )
= sqrt( 0 + 4 )
= sqrt(4)
= 2

Step 3: Determine the equation of the parabola.
The standard form of the equation for a parabola is:
(y - k) = 4a(x - h)^2
where (h, k) is the vertex and 4a is the coefficient that determines the shape of the parabola.

Since the parabola opens either upward or downward due to the directrix being horizontal, we can determine the direction by looking at the distance between the vertex and the focus. If the focus is above the vertex, the parabola opens upward; if the focus is below the vertex, the parabola opens downward.

In this case, since the focus (5,7) is above the vertex (5,5), the parabola opens upward.

Substituting the values into the equation, we have:
(y - 5) = 4a(x - 5)^2

Step 4: Determine the value of 'a'.
Using the distance between the vertex and the focus, we can find the value of 'a'. The value of 'a' is equal to half the distance between the vertex and the focus.

a = 2 / 2
= 1

Final Step: Substitute the value of 'a' into the equation.
(y - 5) = 4(x - 5)^2

Therefore, the equation of the parabola with directrix y = 3 and focus (5,7) is:
(y - 5) = 4(x - 5)^2