Question

Find the endpoints of the latus rectum of the parabola with equation (x+9)^2 = -20(y+5)

Answer 1

The endpoints of the latus rectum for the parabola (x+9)^2 = -20(y+5) are found by first determining the vertex (h, k) and the value of p, then using the formula y = k ± 2p, with h as the x-coordinate. The endpoints are A(-9, 5) and B(-9, -15).

Step-by-step explanation:

To find the endpoints of the latus rectum for the parabola given by the equation (x+9)^2 = -20(y+5), we should follow the standard procedure for parabolas with a vertical axis of symmetry.

The general form of the equation for a parabola with a vertical axis of symmetry is (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, and also from the vertex to the directrix.

Comparing this with the given equation (x+9)^2 = -20(y+5), we can see that h = -9, k = -5, and 4p = -20, so p = -5.

The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry that passes through the focus. Since the length of the latus rectum is 4p, the endpoints of the latus rectum on this parabola are at x = h, and y = k ± 2p.

Plugging in the values of h, k, and p, we get the endpoints A and B of the latus rectum as A(-9, 5) and B(-9, -15).

Answer 2

The end points of the given parabola is (-19, -5) and (1, -5).

Equation of the given parabola is
`\((x + 9)^2 = -20(y + 5)\)`.

To find the endpoints of the latus rectum (also known as the focal width), we need to identify the vertex and the focus of the parabola.

The given equation is in the form:

`\((x - h)^2 = 4a(y - k)\)`,

where (h, k) is the vertex of the parabola and 4a is the focal width.

Comparing this with the given equation,

`\((x + 9)^2 = -20(y + 5)\)`

h = -9 and k = -5

4a = -20

a = -5

Vertex = (h, k) = (-9, -5)

Focal width = 4a = -20

The latus rectum is parallel to the x-axis and has endpoints that are equally distant from the focus (vertex) on either side. The length of the latus rectum is equal to the absolute value of the focal width.

The endpoints of the latus rectum can be found as:

Endpoint 1:

(-9 - 10, -5) = (-19, -5)

Endpoint 2:

(-9 + 10, -5) = (1, -5)

Therefore, the endpoints of the latus rectum of the given parabola are (-19, -5) and (1, -5).