Question

Justify if the point (6, 5) is on the parabola with a focus at (2, 2), a vertex at (2, 1) and the directrix is on the x-axis (y = 0). Be sure to use words, numbers or both in your justification.

Answer 1

Explanation:

To determine whether the point (6, 5) lies on the parabola with a focus at (2, 2), a vertex at (2, 1), and the directrix on the x-axis (y = 0), we can use the definition of a parabola.

A parabola is the set of all points that are equidistant from the focus and the directrix. This distance is also equal to the distance between the point and the vertex.

Let's first find the equation of the parabola. Since the vertex is (2, 1) and the directrix is the x-axis, the axis of symmetry is the line x = 2. This means that the parabola has the equation:

(y - 1)^2 = 4p(x - 2)

where p is the distance between the vertex and the focus, which we need to find. Since the focus is at (2, 2), which is one unit above the vertex, we know that p = 1/4. Substituting this value into the equation above, we get:

(y - 1)^2 = (x - 2)

Now we can check whether the point (6, 5) satisfies this equation. Plugging in x = 6 and y = 5, we get:

(5 - 1)^2 = (6 - 2)

16 = 4

This equation is clearly false, which means that the point (6, 5) does not lie on the parabola with the given focus, vertex, and directrix. Therefore, we can conclude that the point (6, 5) is not equidistant from the focus and the directrix and therefore does not lie on the parabola

Answer 2

The equation

We can tell that this parabola opens upwards since the directrix lies on the horizontal axis and the vertex is higher (positive p), so we know this is an x² parabola.

The form of parabola equation we will use with the given information will be (x-h)²=4p(y-k), where h,k is the vertex.

p refers to the distance between either the focus and the vertex or the vertex and the directrix. The y value of the focus is 2, and the y value of the vertex is 1. 2-1=1, so p=1.

From the vertex, (2,1), we know that h = 2 and k = 1.

Therefore, the equation is

(x-2)²=4(1)(y-1)

Verifying

From this, we can now plug in 6 and 5 for x and y in the equation respectively to see if that point falls on it.

(6-2)²=4(5-1)

4²=4*4

16=16

This statement is true, so the point (5,6) does fall on the parabola.