   Chapter 10.5, Problem 56E

Chapter
Section
Textbook Problem

(a) Show that the equation of the tangent line to the parabola y2 = 4px at the point (x0, y0) can be written as y 0 y = 2 p ( x + x 0 ) (b) What is the x-intercept of this tangent line? Use this fact to draw the tangent line.

a)

To determine

To show: The equation of the tangent line to the parabola y2=4px at the point (x0,y0) can be rewritten as y0y=2p(x+x0) .

Explanation

Given:

The equation to parabola is y2=4px and the tangent line at the point (x0,y0) .

Calculation:

The equation of the parabola is as below.

y2=4px

Differentiate the above equation with respective to x for both the side.

y2dydx=4pxdydxy2dydx=4p

Use the chain in formula in LHS .

y2dxdy×dydx=4p2y×dydx=4pdydx=4p2y

dydx=2pyy'=2py

Then the slope of the tangent is (dydx)(x0,y0)=2py0 .

Substitute the above slope 2py0 in the equation of the line and pass through the point (x0,y0)

b)

To determine

To find: The x intercept of the tangent line in part (a) and draw the tangent line.

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