Question

Write the equation for a parabola with a focus at (-2, 5) and a directrix at x = 3.

Answer 1

Solid chance this is way above your knowledge level



A parabola is a curve in the shape of a U that is defined as the set of all points that are equidistant to a fixed point (called the focus) and a fixed line (called the directrix).

To write the equation of a parabola with a focus at (-2, 5) and a directrix at x = 3, we can use the standard form of the equation of a parabola, which is:

y = (1/(4f))x^2 + k

Where f is the distance between the focus and the vertex (the point where the parabola changes direction), and k is a constant that determines the position of the parabola along the y-axis.

To find the value of f, we can use the distance formula:

f = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Where (x1, y1) is the coordinate of the focus and (x2, y2) is the coordinate of the vertex.

Since the focus is at (-2, 5) and the directrix is at x = 3, we can use the y-coordinate of the focus as the y-coordinate of the vertex, and the x-coordinate of the directrix as the x-coordinate of the vertex. Therefore, the coordinate of the vertex is (3, 5).

Substituting these values into the distance formula, we get:

f = sqrt((3 - (-2))^2 + (5 - 5)^2)

= sqrt((5)^2 + (0)^2)

= sqrt(25)

= 5

Now that we have the value of f, we can substitute it into the standard form of the equation of a parabola to get:

y = (1/(4*5))x^2 + k

= (1/20)x^2 + k

This is the equation for a parabola with a focus at (-2, 5) and a directrix at x = 3. The constant k determines the position of the parabola along the y-axis.

Answer 2

First, we must find the distance from the focus to the vertex and the vertex to the directrix. There is a special property about parabolas: any point (x, y) on the parabola to the focus is equidistant from that point (x, y) to the directrix. Therefore, this means that from the vertex to the focus and from the vertex to the directrix, the lengths to these two features are equidistant. Generally, this equal distance is called “p.”

Let’s find “p:”

We know the parabola is opens to the left because the directrix is always below the vertex. So, we must find the distance between the two x-coordinates of -2 and 3 and divide that in half

-2+3/2=.5

Because the directrix and focus lie on the same y-coordinate, we can use that coordinate to determine the y-coordinate of the vertex. In this case, y=5.

This means the vertex is at (.5, 5) or (1/2, 5).

Now, we must determine how wide or stretched the parabola is, which is known as the “a” value in a(x-h)^2+k

The formula for a is:

a=1/4p

We know that p=.5

a=1/4(.5)

a=1/2

So, we have all the information to come the equation:

f(x)=1/2(x-1/2)^2+5

However, the function is inverted, so it is reflected across the line y=x. This means we must swap all x and y coordinates:

x=1/2(y-5)^2+1/2 is the equation