what is the standard equation of the following parabolas?!
1. passes through (-5,1)
2. Direction 2x-9=0
3.E(4,2) as end of LR
4. Focus at (3,0)
5. LR=10
pls answer
Sure, I'd be happy to help you find the standard equations of the given parabolas.
1. To find the equation of a parabola that passes through a specific point, we need to use the general form of the equation of a parabola: y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola. Let's substitute the given point (-5,1) into the equation and solve for a.
1 = a(-5-h)^2 + k
Since the equation only contains the variable a, we can solve for a by substituting the coordinates of the point and the vertex into the equation and solve for a:
1 = a(-5-h)^2 + k
1 = a(-5-h)^2 + k
1 = a(-5+5)^2 + 1
1 = a(0)^2 + 1
1 = 0 + 1
1 = 1
Now that we know the value of a is 1, the equation of the parabola is:
y = (x-h)^2 + k
Therefore, the standard equation of the parabola that passes through (-5,1) is: y = (x+5)^2 + 1.
2. To find the equation of a parabola given its directrix and the equation of the line representing its axis of symmetry, we can use the standard form of the equation of a parabola:
y = a(x-h)^2 + k
From the given information that the equation of the line representing the axis of symmetry is 2x - 9 = 0, we can identify that the vertex of the parabola is on this line.
To find the vertex, we set the equation of the line equal to zero and solve for x:
2x - 9 = 0
2x = 9
x = 9/2
Therefore, the vertex is (9/2, k).
Now, we need to find the value of a. The distance between the vertex and the directrix is equal to the distance between the vertex and the focus. The equation of the directrix is x = 9/2, so the distance from the vertex to the directrix is 9/2 - 9/2 = 0.
The equation of the directrix provides the value of k. Since the focus lies on the positive side of the vertex, the value of k is negative. Therefore, k = -9/2.
Now, we can substitute the coordinates of the vertex and the value of k into the equation to solve for a:
0 = a(0 - 9/2)^2 - 9/2
Simplifying:
0 = a(-9/2)^2 - 9/2
0 = a(81/4) - 9/2
0 = 81a/4 - 9/2
To obtain a common denominator:
0 = 81a/4 - 18/4
0 = (81a - 18)/4
Since the denominator is 4, the numerator must be equal to zero for the equation to hold.
81a - 18 = 0
81a = 18
a = 18/81
a = 2/9
Therefore, the value of a is 2/9, and the equation of the parabola is:
y = (2/9)(x - 9/2)^2 - 9/2
3. To find the equation of a parabola given its endpoint and the length of its latus rectum (LR), we need to know the coordinates of the vertex. Unfortunately, this information is not provided. Without the vertex, we cannot determine the specific equation of the parabola.
4. To find the equation of a parabola given its focus, we need to know the coordinates of the vertex as well. Unfortunately, this information is not provided. Without the vertex, we cannot determine the specific equation of the parabola.
5. To find the equation of a parabola given its latus rectum (LR), we need to know the coordinates of the vertex as well. Unfortunately, this information is not provided. Without the vertex, we cannot determine the specific equation of the parabola.
In summary, we were able to find the standard equations for the following parabolas:
1. y = (x+5)^2 + 1
2. y = (2/9)(x - 9/2)^2 - 9/2
We were not able to find the standard equations for the parabolas described in point 3, 4, and 5, as the necessary information (vertex coordinate) was not provided.