Please help me with this problem. I already figured out the shape of the figure (figures) but I can't figure out how to get what a or b is. Thank you so much for your help!
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x² + y = 100
x² = - y + 100
x² = 4(-1/4)(y - 100)
Vertex: (0, 100)
Focus: (0, 399/4)
Directrix: y = 401/4
The required ellipse has foci (0, 0) and (0, 399/4), so its center is (0, 399/8). It has the same directrix at top. The focus distance is 399/8, the distance from center to directrix is 403/8. The semimajor is the geometric mean of 399/8 and 403/8.
semimajor² = (399/8)(403/8) = 160797/64
semiminor² = 160797/64 - (399/8)² = 1596/64
x² / (1596/64) + (y - 399/8)² / (160797/64) = 1
That settles in to this general form:
1612x² + 16y² - 1596y - 399 =0
x² = - y + 100
x² = 4(-1/4)(y - 100)
Vertex: (0, 100)
Focus: (0, 399/4)
Directrix: y = 401/4
The required ellipse has foci (0, 0) and (0, 399/4), so its center is (0, 399/8). It has the same directrix at top. The focus distance is 399/8, the distance from center to directrix is 403/8. The semimajor is the geometric mean of 399/8 and 403/8.
semimajor² = (399/8)(403/8) = 160797/64
semiminor² = 160797/64 - (399/8)² = 1596/64
x² / (1596/64) + (y - 399/8)² / (160797/64) = 1
That settles in to this general form:
1612x² + 16y² - 1596y - 399 =0
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4p [ y-k] = [ x - h ]² is key...you have - [ y - 100] = x ²---> vertex at ( 0 , 100) and p = -1/4
thus a focus is ( 0 , 99.75) as well as (0.0)....thus a = 100 , center is ( 0 , 49.875) , c = 49.875
and a ² = b ² + c ²
thus a focus is ( 0 , 99.75) as well as (0.0)....thus a = 100 , center is ( 0 , 49.875) , c = 49.875
and a ² = b ² + c ²