SOLUTION: Reduce the equation to standard form x^2+4y^2+6x+16y+21=0. Find the center, foci, ends of x and y axes.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Reduce the equation to standard form x^2+4y^2+6x+16y+21=0. Find the center, foci, ends of x and y axes.      Log On


   



Question 1004568: Reduce the equation to standard form x^2+4y^2+6x+16y+21=0. Find the center, foci, ends of x and y axes.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2B4y%5E2%2B6x%2B16y%2B21=0
%28x%5E2%2B6x%29%2B%284y%5E2%2B16y%29%2B21=0...........complete square
%28x%5E2%2B6x%2Bb%5E2%29-b%5E2%2B%284y%5E2%2B16y%2Bb%5E2%29-b%5E2%2B21=0
%28x%5E2%2B6x%2Bb%5E2%29-b%5E2%2B4%28y%5E2%2B4y%2Bb%5E2%29-4b%5E2%2B21=0
for x part we have a=1 and 2ab=6=>2%2A1%2Ab=6=>b=3
for y part we have a=1 and 2ab=4=>2%2A1%2Ab=4=>b=2

%28x%5E2%2B6x%2B3%5E2%29-3%5E2%2B%284y%5E2%2B16y%2B2%5E2%29-4%2A2%5E2%2B21=0
%28x%2B3%29%5E2-9%2B4%28y%2B2%29%5E2-16%2B21=0
%28x%2B3%29%5E2%2B4%28y%2B2%29%5E2-25%2B21=0
%28x%2B3%29%5E2%2B4%28y%2B2%29%5E2-4=0
%28x%2B3%29%5E2%2B4%28y%2B2%29%5E2=4

%28x%2B3%29%5E2%2F4%2B4%28y%2B2%29%5E2%2F4=4%2F4
%28x%2B3%29%5E2%2F4%2B%28y%2B2%29%5E2%2F1=1=>h=-3,k=-2, a=2,b=1
c%5E2=a%5E2-b%5E2
c%5E2=2%5E2-1%5E2
c%5E2=4-1
c%5E2=3
c=sqrt%283%29

the center:(-3,-2 )
semimajor axis length: +2
semiminor axis length: 1
foci: (h-c%2C+-2) and (h%2Bc%2C+-2)
(-3-sqrt%283%29, -2) and (-3%2Bsqrt%283%29, -2)
or (-4.7,+-2) and (-1.3,+-2)
vertices: (h-a, -2) and (h%2Ba, -2)
(-5, -2) and (-1, -2)