Write the equation of the parabola

1.Vertex(2,5) Focus(2,-3)
2.Directrix=x=3/4 Focus(5/4,-3)

For 1. i got(x-2)^2=32(y-5)^2
For 2. i got y=8x^2-20x-61/60

1. V(2,5), F(2,-3).

1/4a = -3-5 = -8.
4a = -1/8.
a = -1/32.

Y = a(x-h)^2 + k.
Y = (-1/32)(x-2)^2 + 5, Vertex Form

2. D(3/4,-3), V(h,-3), F(5/4,-3).

1/2a = 5/4-3/4 = 2/4 = 1/2.
2a = 2, a = 1.

1/4a = 1/(4*1) = 1/4.

h = 3/4 + 1/4a = 3/4 + 1/4 = 4/4 = 1.

X = a(y-k)^2 + h.
X = (y+3)^2 + 1, Vertex form.

X = y^2 + 6y + 10, Standard form.

To find the equation of a parabola given the vertex and either the focus or directrix, you can use the standard form equation for a parabola:

1. For the parabola with vertex (2, 5) and focus (2, -3):
Since the vertex and focus have the same x-coordinate (2 in this case), the parabola opens either upwards or downwards. The vertex form equation for a parabola opening upwards or downwards is given by:
(x - h)^2 = 4p(y - k)
where (h, k) represents the vertex and p is the distance between the vertex and focus (or vertex and directrix).

In this case, the vertex is (2, 5) and the focus is (2, -3). The distance between the vertex and the focus is |-3 - 5| = 8. Since the parabola opens upwards/downwards (not sideways), we can use the equation:
(x - 2)^2 = 4p(y - 5)
Substituting the value of p, we get:
(x - 2)^2 = 4(8)(y - 5)
Simplifying further, we have:
(x - 2)^2 = 32(y - 5)
So, the equation of the parabola is (x - 2)^2 = 32(y - 5).

2. For the parabola with directrix x = 3/4 and focus (5/4, -3):
Since the directrix is a vertical line, the parabola opens either leftwards or rightwards. The vertex form equation for a parabola opening leftwards or rightwards is given by:
(y - k)^2 = 4p(x - h)
where (h, k) represents the vertex and p is the distance between the vertex and focus (or vertex and directrix).

In this case, the vertex is not given directly, but we know the directrix is x = 3/4. Since the directrix is a vertical line, the vertex lies on a horizontal line with the y-coordinate being the same as the y-coordinate of the focus (-3 in this case). The distance between the vertex and directrix is the absolute difference of the x-coordinates of the focus and the directrix, which is |(5/4) - (3/4)| = 2/4 = 1/2.

Now, substituting the values into the equation:
(y - (-3))^2 = 4(1/2)(x - h)
Simplifying further, we have:
(y + 3)^2 = 2(x - h)
To find the value of h, we can observe that the vertex lies on the line of symmetry, i.e., the midpoint of the focus and the directrix. The x-coordinate of the midpoint is [(5/4) + (3/4)]/2 = 4/4 = 1. So the vertex is (1, -3). Substituting the values of h and simplifying, we have:
(y + 3)^2 = 2(x - 1)
Expanding the equation further, we get:
y^2 + 6y + 9 = 2x - 2

Rearranging the terms and bringing them to one side, we have:
2x - y^2 - 6y - 11 = 0
So, the equation of the parabola is 2x - y^2 - 6y - 11 = 0.