   Chapter 10.1, Problem 50E

Chapter
Section
Textbook Problem

# Finding Equations of Tangent Lines and Normal Lines In Exercises 49 and 50, find equations fur (a) the tangent lines and (b) the normal lines to the hyperbola for the given value of x. (The normal line at a point is perpendicular to the tangent line at the point.) y 2 4 − x 2 2 = 1 ,   x = 4

(a)

To determine

To-calculate: The of the tangent lines for the hyperbola y24x22=1 at x=4.

Explanation

Given:

The equation of the hyperbola, y24x22=1 and x=4.

Formula used:

Equation of a line passes through (x1,y1) and slope m is (yy1)=m(xx1).

Calculation:

Consider the provided equation of the hyperbola, y24x22=1.

Now, substitute x=4 in the equation of the hyperbola, y24x22=1, to find the coordinate for y.

So,

y24422=1y24=1+8y2=36y=±6

Thus, the two coordinates are (4,6) and (4,6).

Now, differentiate the equation of the hyperbola, y24x22=1 with respect to x.

So,

12ydydxx=0.

dydx=2xy …...…... (1)

Since, dydx=2xy is the slope of the curve at any point on the curve of the hyperbola, x29y2=1

So, the slope of the hyperbola, dydx=2xy at (4,6) is,

dydx=2(4)6=86=43

Now, with the help of the standard equation of the line (yy1)=m(xx1), calculate the equation of the tangent passing through (4,6) and with a slope of 43

(b)

To determine

To-calculate: The equations of the normal lines for the hyperbola y24x22=1 at x=4.

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