   Chapter 10.5, Problem 57E

Chapter
Section
Textbook Problem

Show that the tangent lines to the parabola x2 = 4py drawn from any point on the directrix are perpendicular.

To determine

To show: The tangent lines to the parabola x2=4py drawn from any point on the directrix are perpendicular.

Explanation

Given:

The equation to parabola is x2=4py .

Calculation:

The equation of the parabola is as below.

x2=4py

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

x2dydx=4pydydx2xdydx=4pydydx2x=4py'

2x4p=y'y'=x2p

Then the slope of the tangent is (dydx)(x0,y0)=x2p .

Let rewrite the equation as x02=4py0 .

y0=x024p

Substitute the above slope x02p in the equation of the line and pass through the point (x0,y0) .

yy0xx0=x02p

(yy0)=x2p(xx0) (1)

Substitute the value x024p for y0 in equation (1).

yx024p=x02p(xx0)

Substitute the value (a,p) for (x,y) in above equation

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