   Chapter 2.6, Problem 45E

Chapter
Section
Textbook Problem

# Find an equation of the tangent line to the hyperbola x 2 a 2 − y 2 b 2 = 1 at the point ( x 0 , y 0 ) .

To determine

To find:

Equation of tangent line to hyperbola at the point x0,y0

Explanation

1) Formula:

ddxxn=nxn-1

2) Given:

x2a2-y2b2=1

3) Calculation:

Slope of tangent at a point is the derivative with respect to x at that point, So let us differentiate the curve with respect to x

ddxx2a2-y2b2=1

ddxx2a2-ddxy2b2=ddx1

1a2ddxx2-1b2ddxy2=ddx1

2xa2-2yb2dydx=0

xa2-yb2dydx=0 Thus we have

-yb2dydx=-xa2

dydx=xb2ya2

Slop of tangent at

x0,y0=dydxx0,y0=x0b2y0a2

Slope

m =x0b2y0a2

Use point slope form to obtain an equation of line,

Equation of tangent at x0,y0 is

y-y0y0a2=x0b2(x-x0)

yy0a2-y0

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