Question #270081

find the center,foci,vertices, endpoints of conjugate axis.determine the equation of the asymptotes and sketsch the graph



1. (y+6)²/25-(x-4)²/39=1



2. 9x²+126x-16y²-96y+153=0


1
Expert's answer
2021-11-25T23:48:20-0500

1.


(y+6)225(x4)239=1\dfrac{(y+6)^2}{25}-\dfrac{(x-4)^2}{39}=1

Center: (4,6)(4, -6)

Vertices: (4,11),(4,1)(4, -11), (4, -1)

c2=a2+b2=25+39=64=>c=8c^2=a^2+b^2=25+39=64=>c=8

Foci: (4,14),(4,2)(4,-14), (4, 2)

Endpoints of conjugate axis: (439,6),(4+39,6)(4-\sqrt{39}, -6), (4+\sqrt{39}, -6)


First asymptote: y=53939(x4)6y=-\dfrac{5\sqrt{39}}{39}(x-4)-6


Second asymptote: y=53939(x4)6y=\dfrac{5\sqrt{39}}{39}(x-4)-6




2.


9x2+126x16y296y+153=09x^2+126x-16y^2-96y+153=0

9(x2+14x+49)16(y2+6y+9)=1449(x^2+14x+49)-16(y^2+6y+9)=144

(x+7)216(y+3)29=1\dfrac{(x+7)^2}{16}-\dfrac{(y+3)^2}{9}=1

Center: (7,3)(-7, -3)

Vertices: (11,3),(3,3)(-11, -3), (-3, -3)

c2=a2+b2=16+9=25=>c=5c^2=a^2+b^2=16+9=25=>c=5

Foci: (12,3),(2,3)(-12,-3), (-2,-3)

Endpoints of conjugate axis: (7,6),(7,0)(-7, -6), (-7,0)


First asymptote: y=34x334y=-\dfrac{3}{4}x-\dfrac{33}{4}


Second asymptote: y=34x+94y=\dfrac{3}{4}x+\dfrac{9}{4}





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