   Chapter 10.1, Problem 89E

Chapter
Section
Textbook Problem

# Tangent Line Show that equation of the tangent line to x 2 a 2 − y 2 b 2 = 1 at the point ( x 0 ,   y 0 )   is   ( x 0 a 2 ) x − ( y 0 b 2 ) = 1 .

To determine

To Prove: The equation of tangent line to the hyperbola x2a2y2b2=1 is x0a2xy0b2y=1 at the point (x0,y0).

Explanation

Given:

The equation of hyperbola, x2a2y2b2=1.

Formula used:

Straight line equation in slope and point, yy0=m(xx0).

Proof:

The equation of hyperbola is,

x2a2y2b2=1

Differentiating the above equation,

ddx(x2a2y2b2)=d(1)dx2xa22yb2(dydx)=0dydx=b2xa2y

Slope at a point (x0,y0).

dydx=b2x0a2y0

The equation of the straight line passing through the point (x0,y0),

(yy0)=x0b2y0a2(xx0)y0a2y

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