Question #223141

Given the conic section -3.6x2+1.6y2+72x+6.4y = 58/5

a) State the nature of the conic section

b) Find the characteristic elements of the conic section

c) Sketch the conic section indicating all the elements.


1
Expert's answer
2021-09-28T15:01:50-0400

a)


3.6x2+1.6y2+72x+6.4y=58/5-3.6x^2+1.6y^2+72x+6.4y = 58/5

3.6(x220x+100)+360+1.6(y2+4y+4)6.4=11.6-3.6(x^2-20x+100)+360+1.6(y^2+4y+4)-6.4=11.6

3.6(x10)21.6(y+2)2=3423.6(x-10)^2-1.6(y+2)^2=342

(x10)295(y+2)2213.75=1\dfrac{(x-10)^2}{95}-\dfrac{(y+2)^2}{213.75}=1

A conic is a hyperbola. Standard form


(x10)295(y+2)2213.75=1\dfrac{(x-10)^2}{95}-\dfrac{(y+2)^2}{213.75}=1

Horizontal hyperbola.

b)

h=10,k=2,a=95,b=213.75=1.595h=10, k=-2, a=\sqrt{95}, b=\sqrt{213.75}=1.5\sqrt{95}

c2=a2+b2=95+213.75=308.75,c=308.75c^2=a^2+b^2=95+213.75=308.75, c=\sqrt{308.75}

Center: (h,k)=(10,2)(h, k)=(10,-2)

Vertices: (h±a,k),(1095,2),(10+95,2)(h\pm a,k), (10-\sqrt{95}, -2), (10+\sqrt{95}, -2)

Covertices: (h,k±b),(10,21.595),(10,2+1.595)(h, k\pm b), (10, -2-1.5\sqrt{95}), (10, -2+1.5\sqrt{95})

Foci: (h±c,k),(10308.75,2),(10+308.75,2)(h\pm c,k), (10-\sqrt{308.75}, -2), (10+\sqrt{308.75}, -2)

The equations of the asymptotes are 

y=ba(xh)+k,y=ba(xh)+ky=\dfrac{b}{a}(x-h)+k,y=-\dfrac{b}{a}(x-h)+ky=1.5(x10)2,y=1.5(x10)2y=1.5(x-10)-2,y=-1.5(x-10)-2

x=0,(010)295(y+2)2213.75=1x=0,\dfrac{(0-10)^2}{95}-\dfrac{(y+2)^2}{213.75}=1


(y+2)2=11.24(y+2)^2=11.24

y=2±1.55y=-2\pm1.5\sqrt{5}

(0,21.55),(0,2+1.55)(0, -2-1.5\sqrt{5}), (0, -2+1.5\sqrt{5})



y=0,(x10)295(0+2)2213.75=1y=0,\dfrac{(x-10)^2}{95}-\dfrac{(0+2)^2}{213.75}=1


(x10)2=8399(x-10)^2=\dfrac{839}{9}

x=10±8393x=10\pm\dfrac{\sqrt{839}}{3}

(108713,0),(10+8713,0)(10-\dfrac{\sqrt{871}}{3},0), (10+\dfrac{\sqrt{871}}{3}, 0)

c)

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