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1. For the equation of a hyperbola, identify the asymptotes:
-4x² + 40x+25y2-100y +100=0.

1 Answer

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Answer:

Equations of asymptotes are
\displaystyle y=(2)/(5)x+4 and
\displaystyle y=-(2)/(5)x

Explanation:

Determine the equation of the hyperbola by completing the square:


-4x^2+40x+25y^2-100y+100=0\\\\-4(x^2+10x)+25(y^2-4y+4)=0\\\\-4(x^2+10x+25)+25(y-2)^2=0-4(25)\\\\-4(x+5)^2+25(y-2)^2=-100\\\\((x+5)^2)/(25)-((y-2)^2)/(4)=1

Therefore,
a=5 and
b=2 with center
(h,k)=(-5,2). The asymptotes of the hyperbola can be determined with the equation
\displaystyle y=\pm(b)/(a)(x-h)+k:


\displaystyle y=\pm(b)/(a)(x-h)+k\\\\y=\pm(2)/(5)(x-(-5))+2\\\\y=\pm(2)/(5)(x+5)+2

The first equation is:


\displaystyle y=(2)/(5)(x+5)+2\\\\y=(2)/(5)x+2+2\\\\y=(2)/(5)x+4

The second equation is:


\displaystyle y=-(2)/(5)(x+5)+2\\\\y=-(2)/(5)x-2+2\\\\y=-(2)/(5)x

I've attached a graph to provide a visual of everything. Hopefully, it helps!

1. For the equation of a hyperbola, identify the asymptotes: -4x² + 40x+25y2-100y-example-1
User Praveen Kishor
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