1)Write an equation for the parabola with focus (4,0) and directrix y=2
4c(y - k) = (x - h)^2
4(-1)(y - 1) = (x - 4)^2
-4y + 4 = (x - 4)^2
-4y = (x - 4)^2 - 4
y = (-1/4)(x - 4)^2 + 1
2)Write an equation for the ellipse with end points of the major axis at (7,1) and (-7,1) and end points of the
minor axis at (0,5) and (0,-3).
Major axis is 2a.
2a = 14
a = 7
Minor axis is 2b.
2b = 8
b = 4
(1/7^2)(x - 0)^2 + (1/4^2)(y - 1)^2 = 1
(1/49)x^2 + (1/16)(y - 1)^2 = 1
To write the equation for a parabola with focus (4,0) and directrix y=2, we can use the formula:
4c(y - k) = (x - h)^2
where (h,k) is the vertex and c is the distance from the vertex to the focus (or directrix).
In this case, the vertex is at (h,k) = (4,1) since it lies halfway between the focus and directrix.
The distance from the vertex to the focus, c, is 1, as it is the difference between the y-coordinate of the vertex and the y-coordinate of the focus.
Plugging in the values, the equation becomes:
4(1)(y - 1) = (x - 4)^2
Expanding and simplifying, we get:
4y - 4 = (x - 4)^2
To isolate y, we divide both sides by 4:
y - 1 = (1/4)(x - 4)^2
Adding 1 to both sides, we have the equation:
y = (1/4)(x - 4)^2 + 1
Therefore, the equation of the parabola is y = (1/4)(x - 4)^2 + 1.
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To write the equation for an ellipse with endpoints of the major axis at (7,1) and (-7,1) and endpoints of the minor axis at (0,5) and (0,-3), we can use the formula:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
where (h,k) is the center of the ellipse, and a and b are half the lengths of the major and minor axes, respectively.
In this case, the center of the ellipse is (h,k) = (0,1) since it lies halfway between the endpoints.
The length of the major axis is 2a, and we can determine it based on the given endpoints:
2a = distance between (-7,1) and (7,1) = 14
Dividing by 2, we find a = 7.
Similarly, the length of the minor axis is 2b, and we can determine it based on the given endpoints:
2b = distance between (0,5) and (0,-3) = 8
Dividing by 2, we find b = 4.
Plugging in these values, the equation becomes:
((x - 0)^2 / 7^2) + ((y - 1)^2 / 4^2) = 1
Simplifying, we have:
(1/49)x^2 + (1/16)(y - 1)^2 = 1
Therefore, the equation of the ellipse is (1/49)x^2 + (1/16)(y - 1)^2 = 1.