SOLUTION: (Ellipse) 25x^2 + 64y^2 - 400x - 600y=0 Find its standard equation, foci, vertices and extemeties of the minor axis.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: (Ellipse) 25x^2 + 64y^2 - 400x - 600y=0 Find its standard equation, foci, vertices and extemeties of the minor axis.      Log On


   

Question 987973: (Ellipse)
25x^2 + 64y^2 - 400x - 600y=0
Find its standard equation, foci, vertices and extemeties of the minor axis.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
25x%5E2+%2B+64y%5E2+-+400x+-+600y=0
%2825x%5E2++-+400x%29+%2B+%2864y%5E2-+600y%29=0.........complete square
25%28x%5E2-16x%29+%2B+64%28y%5E2-%2875%2F8%29y%29=0
25%28x%5E2+-16x%2B_%29+-25%2A_%2B+64%28y%5E2-%2875%2F8%29y%2B_%29-8%2A_=0
.........since 2ab=2%2A1%2Ab=75%2F8...b=75%2F16
25%28x+-8%29%5E2+-25%2A64%2B+64%28y-+75%2F16%29%5E2-64%2A%2875%2F16%29%5E2=0
25%28x++-+8%29%5E2+-1600%2B+64%28y-+75%2F16%29%5E2-5625%2F4=0
25%28x++-+8%29%5E2+%2B+64%28y-+75%2F16%29%5E2-12025%2F4=0....multiply by 4
100%28x++-+8%29%5E2+%2B256%28y-+75%2F16%29%5E2=12025....divide by 12025
100%28x++-+8%29%5E2+%2F12025%2B256%28y-+75%2F16%29%5E2%2F12025=12025%2F12025
%284%2F481+%29%28x-8%29%5E2%2B%28256%2F12025%29%28y-+75%2F16%29%5E2=1
%28x-8%29%5E2%2F%28481%2F4+%29%2B%28y-+75%2F16%29%5E2%2F%2812025%2F256%29=1
now we have a standard form and we know that:
a%5E2=481%2F4 => a=sqrt%28481%29%2F2
b%5E2=12025%2F256 => b=5sqrt%28481%29%2F16
we can calculate c using formula c%5E2=a%5E2-b%5E2=>c=sqrt%28a%5E2-b%5E2%29
c+=+sqrt%28481%2F4-12025%2F256%29
c+=+13sqrt%28111%29%2F16
we also know that
h=8
k=75%2F16
=>so, the center is at (8, 75%2F16), approximately at (8, 4.7)
foci:(8-13sqrt%28111%29%2F16, 75%2F16) or (8%2B13sqrt%28111%29%2F16, 75%2F16), approximately at (-0.6, 4.7) or (16.6, 4.7)
vertices: (8-sqrt%28481%29%2F2, 75%2F16) or (8%2Bsqrt%28481%29%2F2, 75%2F16), approximately at (-2.97, 4.7) or (18.97, 4.7)
semi-major axis length: sqrt%28481%29%2F2~~10.97
semi-minor axis length: (5sqrt%28481%29%29%2F16~~6.9
eccentricity | sqrt%2839%29%2F8~~0.8