Find the equation of the parabola through (-15/4 ,2) and (0,-1) whose axis is parallel to the x-axis and the latus rectum equal to 4
(y-3)SQUARE = 4(X+4) AND
(Y+2)SQUARE = -4(X-1/4)
Graph the parabola x = 2y^2+8y+11. Find the vertex , an equation of the axis, the focus, and the endpoints of the latus rectum
To find the equation of the parabola, we can start by finding the vertex and the focus of the parabola.
Since the axis of the parabola is parallel to the x-axis, we know that the parabola has the equation of the form: y = a(x - h)^2 + k. We need to find the values of a, h, and k.
Step 1: Finding the coordinates of the vertex (h, k):
Since the axis is parallel to the x-axis, the vertex of the parabola lies midway between the two given points. Therefore, the x-coordinate of the vertex is the average of the x-coordinates of the given points, and the y-coordinate is the average of the y-coordinates.
Vertex formula: (h, k) = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the given points:
Vertex formula: (h, k) = ((-15/4 + 0)/2, (2 - 1)/2)
= (-15/8, 1/2)
So, the vertex of the parabola is (-15/8, 1/2).
Step 2: Finding the value of 'a':
Since the latus rectum is equal to 4, we know that the distance from the vertex to the focus (p) is equal to half the latus rectum.
Latus Rectum Length formula: 4 = 4a/p
Substituting the coordinates of the vertex (-15/8, 1/2):
4 = 4a/p
4 = 4a/(1/2)
4 = 8a
a = 1/2
So, the value of 'a' is 1/2.
Step 3: Finding the focus of the parabola:
The focus of the parabola is given by the formula:
(h + p, k)
Substituting the coordinates of the vertex (-15/8, 1/2):
(h + p, k) = (-15/8 + p, 1/2)
Since the latus rectum length (p) is equal to 4, we know that the distance from the vertex to the focus is 4:
Distance formula: 4 = sqrt(((-15/8 + p) - (-15/8))^2 + ((1/2) - (1/2))^2)
Simplifying the equation, we get: √(128p^2 + 64) = 4
Squaring both sides: 128p^2 + 64 = 16
Simplifying the equation, we get: 128p^2 = -48
p^2 = -48/128
p^2 = -3/8
Since the parabola faces upwards, p = sqrt(-3/8)
So, the focus of the parabola is (-15/8 + sqrt(-3/8), 1/2) or (-15/8 - sqrt(-3/8), 1/2).
The equation of the parabola is y = a(x - h)^2 + k.
Substituting the values of 'a', 'h', and 'k' in the equation, we get:
y = (1/2)(x - (-15/8))^2 + 1/2
y = (1/2)(x + 15/8)^2 + 1/2
Hence, the equation of the parabola passing through the points (-15/4, 2) and (0, -1), with an axis parallel to the x-axis and a latus rectum length of 4 is: y = (1/2)(x + 15/8)^2 + 1/2.
To find the equation of the parabola, we can start by determining the vertex and the focus of the parabola.
1. Finding the vertex:
Since the axis of the parabola is parallel to the x-axis, the vertex takes the form (h, k). We can find the x-coordinate of the vertex, h, by averaging the x-coordinates of the given points: (-15/4 + 0) / 2 = -15/8.
To find the y-coordinate of the vertex, k, we average the y-coordinates of the given points: (2 - 1) / 2 = 1/2.
Thus, the vertex of the parabola is (-15/8, 1/2).
2. Finding the focus:
The latus rectum is the segment passing through the focus, perpendicular to the axis, and with a length equal to 4. Since the latus rectum length is 4, the distance from the vertex to the focus (called the focal length) is also 4/2 = 2.
Given that the axis is parallel to the x-axis, the focus will be vertically above or below the vertex. Thus, the focus will have the coordinates (-15/8, 1/2 + 2) = (-15/8 , 2.5).
3. Determining the equation of the parabola:
Since the axis is parallel to the x-axis, the equation of the parabola takes the form: (y - k) = a(x - h)^2.
To find the value of 'a', we can use the known length of the latus rectum:
The equation for the latus rectum is 4a = 4.
Simplifying, we find that a = 1.
Now we have:
(y - 1/2) = (x + 15/8)^2.
Therefore, the equation of the parabola is:
y = (x + 15/8)^2 + 1/2