SOLUTION: how to find a b and c, eccentricity, directrix, sf, gf endpoints of LR, LR B, V(vertex) and foci 9x^2 + 4y^2 - 24y -72 + 144 = 0 (ellipse)

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: how to find a b and c, eccentricity, directrix, sf, gf endpoints of LR, LR B, V(vertex) and foci 9x^2 + 4y^2 - 24y -72 + 144 = 0 (ellipse)      Log On


   

Question 1121763: how to find a b and c, eccentricity, directrix, sf, gf endpoints of LR, LR B, V(vertex) and foci
9x^2 + 4y^2 - 24y -72 + 144 = 0
(ellipse)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

9x%5E2+%2B+4y%5E2+-+24y+-72++%2B+144+=+0+...-> I guess you got 72x
9x%5E2+%2B+4y%5E2+-+24y+-72x+%2B+144+=+0+
%289x%5E2-72x+%29%2B+%284y%5E2+-+24y+%29+%2B+144+=+0+
9%28x%5E2-8x+%29%2B4+%28y%5E2+-+6y+%29+%2B+144+=+0+
9%28x%5E2-8x%2Bb%5E2+%29+-9b%5E2%2B4+%28y%5E2+-+6y+%2Bb%5E2%29+-4b%5E2%2B+144+=+0+
9%28x%5E2-8x%2B4%5E2+%29+-9%2A4%5E2%2B4+%28y%5E2+-+6y+%2B3%5E2%29+-4%2A3%5E2%2B+144+=+0+
9%28x-4%29%5E2++-9%2A16%2B4+%28y+-+3%29%5E2+-4%2A9%2B+144+=+0+
9%28x-4%29%5E2++-144%2B4+%28y+-+3%29%5E2+-36%2B+144+=+0+
9%28x-4%29%5E2+%2B4+%28y+-+3%29%5E2+-36+=+0+
9%28x-4%29%5E2+%2B4+%28y+-+3%29%5E2+=+36+
9%28x-4%29%5E2%2F+36+%2B4+%28y+-+3%29%5E2+%2F36+=+36+%2F36+
%28x-4%29%5E2%2F+4+%2B%28y+-+3%29%5E2+%2F9=1
major axis is vertical, so compare to %28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%2Fa%5E2+=1, you see that
h=4
k=3
b%5E2=4=>b=2
a%5E2=9=>a=3
eccentricity : c%2Fa
first find c:
c%5E2=a%5E2-b%5E2
c%5E2=3%5E2-2%5E2
c%5E2=9-4
c%5E2=5
c=sqrt%285%29

eccentricity: sqrt%285%29%2F3 ≈ 0.76
center: (4, 3)
foci : (4, 3+-+sqrt%285%29) and (4, 3+%2B+sqrt%285%29)
or ≈ (4, 0.76) and (4, 5.24)
vertices:(4, 0) and (4,+6)
covertices: (6, 3) and (2, 3)