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Question 971565: Identify the conic section. if it is a parabola, give the vertex. if it is a circle, give the center and radius. if it is an ellipse or hyperbola, give the center and foci.
a) 6x^2 - 5y^2 + 12x - 10y - 3 = 0
b) 4x^2 + 3y^2 - 8x + 6y = 2
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Identify the conic section. if it is a parabola, give the vertex. if it is a circle, give the center and radius. if it is an ellipse or hyperbola, give the center and foci.
a) 6x^2 - 5y^2 + 12x - 10y - 3 = 0
6x^2+ 12x - 5y^2 - 10y=3
complete the square:
6(x^2+2x+1) - 5( y^2+2y+1)=3+6-5
6(x+1)^2-5((y+1)^2=4
This is an equation of a hyperbola with horizontal transverse axis
Its standard form of equation: 
center: (-1, -1)
a^2=4/6=2/3
b^2=4/5
foci:
c^2=a^2+b^2=2/3+4/5=10/15+12/15=22/15
c=√(22/15)≈1.2
foci: (-1±c,-1)=(-1±1.2,-1)+(-2.2,-1) and (0.2, -1)
..
b) 4x^2 + 3y^2 - 8x + 6y = 2
4x^2- 8x + 3y^2 + 6y = 2
complete the square:
4(x^2- 2x+1) + 3(y^2 + 2y+1) = 2+4+3
4(x-1)^2+3(y+1)^2=9
This is an equation of an ellipse with vertical major axis
Its standard form of equation: 
center: (1,-1)
a^2=3
b^2=(9/4)
c^2=a^2-b^2=9-9/4=36/4-9/4=27/4
c=√(27/4)=2.6
foci: (1,-1±c)=(1.-1±2.6)=(1,-3.6) and (1,1.6)
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