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
And write the equation of the ellipse
Foci(3,1)and(7,1)
Major axis length=10
I got (x-5)^2/5+(y-1)^2/0=1
as a quick check, go to wolframalpha.com and type in
parabola 8x^2-20x-61/60
It will graph it and describe its properties. Same for the ellipse.
As for #1, you must see that it cannot be a parabola. Not with both x^2 and y^2.
That is not a parabola!
(y-k) should not be squared
should be upside down parabola, sheds water
form
(x-h)^2 = 4 a (y-k)
(x-2)^2 = 4a (y-5)
a = vertex to focus = 8
(x-2)^2 = -32(y-5)
- 32 because sheds water. as x gets big, y goes to - infinity
To find the equation of a parabola given its vertex and focus, we need to know whether the parabola opens upward or downward.
For 1. Vertex(2,5) and Focus(2,-3)
Since the y-coordinate of the focus is smaller than the y-coordinate of the vertex, we know that the parabola opens downward.
The equation of a parabola with vertex (h, k) and focus (h, k + p) where p is the distance between the vertex and the focus, is given by:
(x - h)^2 = -4p(y - k)
In this case, the vertex is (2, 5) and the focus is (2, -3). The distance between the vertex and focus is 8 units (p = -8), so we can substitute these values into the equation:
(x - 2)^2 = -4(-8)(y - 5)
(x - 2)^2 = 32(y - 5)
So, the equation of the parabola is (x - 2)^2 = 32(y - 5)^2.
For 2. Directrix x = 3/4 and Focus (5/4, -3)
Since the x-coordinate of the directrix is smaller than the x-coordinate of the focus, we know that the parabola opens to the right.
The equation of a parabola with vertex (h, k) and focus (h + p, k) where p is the distance between the vertex and the focus, is given by:
(y - k)^2 = 4p(x - h)
In this case, the vertex is (0, 0) as it is not given, the focus is (5/4, -3), and the distance between the vertex and focus is 5/4 units (p = 5/4), we can substitute these values into the equation:
(y - 0)^2 = 4(5/4)(x - 0)
y^2 = 20(x - 0)
y^2 = 20x
So, the equation of the parabola is y^2 = 20x.
Therefore, the final equations are:
1. (x - 2)^2 = 32(y - 5)^2
2. y^2 = 20x