a)
−3.6x2+1.6y2+72x+6.4y=58/5
−3.6(x2−20x+100)+360+1.6(y2+4y+4)−6.4=11.6
3.6(x−10)2−1.6(y+2)2=342
95(x−10)2−213.75(y+2)2=1 A conic is a hyperbola. Standard form
95(x−10)2−213.75(y+2)2=1 Horizontal hyperbola.
b)
h=10,k=−2,a=95,b=213.75=1.595
c2=a2+b2=95+213.75=308.75,c=308.75Center: (h,k)=(10,−2)
Vertices: (h±a,k),(10−95,−2),(10+95,−2)
Covertices: (h,k±b),(10,−2−1.595),(10,−2+1.595)
Foci: (h±c,k),(10−308.75,−2),(10+308.75,−2)
The equations of the asymptotes are
y=ab(x−h)+k,y=−ab(x−h)+ky=1.5(x−10)−2,y=−1.5(x−10)−2
x=0,95(0−10)2−213.75(y+2)2=1
(y+2)2=11.24
y=−2±1.55
(0,−2−1.55),(0,−2+1.55)
y=0,95(x−10)2−213.75(0+2)2=1
(x−10)2=9839
x=10±3839
(10−3871,0),(10+3871,0)
c)
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