Question 578884
solve and graph  
25x^2-16y^2-100x-96y-444=0
complete the square
25(x^2-4x+4)-16(y^2+6y+9)=444+100-144=400
25(x-2)^2-16(y+3)^2=400
divide by 400=
(x-2)^2/16-(y+3)^2/25=1
This equation is that of a hyperbola with horizontal transverse axis of the standard form: (x-h)^2/a^2-(y-k)^2/b^2=1, (h,k) being the (x,y) coordinates of the center.
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For given equation:
center: (2,-3)
a^2=16
a=4
vertices: (2±a,-3)=(2±4,-3)=(-2,-3) and (6,-3)
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b^2=25
b=5
length of conjugate axis=2b=10
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slope of asymptotes=±b/a=±5/4
Asymptotes are straight lines which go thru the center
Equations for asymptotes:
y=(5/4)x+b
solving for b
-3=(5/4)*2+b
b=-11/2=-5.5
equation:
y=5x/4-5.5
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y=(-5/4)x+b
solving for b
-3=(-5/4)*2+b
b=-.5
equation:
y=-5x/4-.5
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See graph below:
y=±(25(x-2)^2/16-25)^.5-3
{{{ graph( 300, 300, -10, 10, -10, 10,(25(x-2)^2/16-25)^.5-3,-(25(x-2)^2/16-25)^.5-3,5x/4-5.5,-5x/4-.5) }}}