Find the equation of the parabola with latus rectum joining the points (3,5) and (3,-3)
To find the equation of the parabola with the given latus rectum joining the points (3,5) and (3,-3), we need to determine the coordinates of the vertex and the value of the p parameter.
Step 1: Find the coordinates of the vertex:
The vertex of the parabola is the midpoint of the latus rectum.
The midpoint formula is given by:
x-coordinate of the vertex = (x1 + x2) / 2
y-coordinate of the vertex = (y1 + y2) / 2
Using the coordinates of the given points (3,5) and (3,-3):
x-coordinate of the vertex = (3 + 3) / 2 = 6 / 2 = 3
y-coordinate of the vertex = (5 + (-3)) / 2 = 2 / 2 = 1
Therefore, the coordinates of the vertex are (3,1).
Step 2: Find the value of p:
The distance between the vertex and either of the two endpoints of the latus rectum is equal to the value of p. The distance formula is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of the vertex (3,1) and one of the endpoints (3,5):
Distance = √((3 - 3)^2 + (5 - 1)^2) = √(0 + 16) = √16 = 4
Therefore, the value of p is 4.
Step 3: Write the equation in vertex form:
The general equation for a parabola in vertex form is:
(y - k) = 4p(x - h)^2
Where (h,k) represents the vertex and p is the distance from the vertex to the focus.
Using the coordinates of the vertex (3,1) and the value of p = 4:
(y - 1) = 4 * 4 (x - 3)^2
(y - 1) = 16 (x - 3)^2
Thus, the equation of the parabola with the given latus rectum is (y - 1) = 16(x - 3)^2.
To find the equation of a parabola with its latus rectum joining two given points, we need to first determine the vertex and the focal length.
The latus rectum is a line segment perpendicular to the axis of symmetry and passes through the focus. The length of the latus rectum is equal to the distance between the two points.
Given that one point is (3, 5) and the other point is (3, -3), the latus rectum has a length of 8 units.
Since the two given points have the same x-coordinate (3), we can conclude that the vertex of the parabola is at the midway point between the two points on the y-axis. Thus, the vertex is (3, 1).
To find the focal length, we need to compute half the length of the latus rectum. In this case, the focal length is 8/2 = 4 units.
Now, we have the vertex (h, k) = (3, 1) and the focal length (p) = 4 units.
The equation of a parabola with a vertical axis of symmetry can be written as:
(y - k) = (1/4p)(x - h)^2
Substituting the values, we get:
(y - 1) = (1/4*4)(x - 3)^2
Simplifying:
(y - 1) = (1/16)(x - 3)^2
Multiplying both sides by 16 to eliminate the denominator:
16(y - 1) = (x - 3)^2
Expanding the equation:
16y - 16 = x^2 - 6x + 9
Rearranging the terms:
x^2 - 6x + 16y - 25 = 0
Hence, the equation of the parabola with the given latus rectum is x^2 - 6x + 16y - 25 = 0.
you have the length of the L.R. = 8
The focus is (3,1)
Recall that the parabola
y^2 = 4px
has latus rectum 4p. The vertex is 2p from the focus.
So, your parabola is
(y-1)^2 = 8(x-1)
see
http://www.wolframalpha.com/input/?i=parabola+%28y-1%29^2+%3D+8%28x-1%29