write an equation for the conic section
Ellipse with vertices at (-2,2)and (4,2) and covertices at (1,1) and (1,3)
the centre is the midpoint along the major axis, or (1,2)
(notice (1,2) is also the midpoint between (1,1) and (1,3))
so a = 3, and b=1
equation:
(x-1)^2 /9 + (y-2)^2 /1 = 1
To write the equation for the ellipse with the given vertices and covertices, you can utilize the standard form equation for an ellipse, which is:
((x-h)^2 / a^2) + ((y-k)^2 / b^2) = 1,
where (h, k) represents the center of the ellipse, 'a' is the distance from the center to the vertices, and 'b' is the distance from the center to the covertices.
First, find the center of the ellipse by taking the average of the x-coordinates of the vertices and the average of the y-coordinates of the covertices:
Center = ((-2 + 4) / 2, (1 + 3) / 2) = (1, 2).
Next, find the values of 'a' and 'b' by considering the distance between the center and the vertices or covertices:
a = distance from the center to the vertices = 4 - 1 = 3,
b = distance from the center to the covertices = 3 - 2 = 1.
Now we can substitute these values into the standard form equation to obtain the equation of the ellipse:
((x - 1)^2 / 3^2) + ((y - 2)^2 / 1^2) = 1.
Simplifying the equation gives the final result:
(x - 1)^2 / 9 + (y - 2)^2 = 1.