   Chapter 10.1, Problem 68E

Chapter
Section
Textbook Problem

# Proof(a) Prove that if any two tangent lines to a parabola intersect at right angles, then their point of intersection must lie on the directrix.(b) Demonstrate the result of pan (a) by showing that thetangent lines to the parabola x 2 − 4 x − 4 y + 8 = 0 at thepoints ( − 2 , 5 ) and ( 3 , 5 4 ) intersect at right angles and that their point of intersection lies on the directrix.

64.a)

To determine

To Prove: If two tangent lines to a parabola intersect at right angles then the point of intersection must lie on the directrix of parabola.

Explanation

Given: Two lines of a parabola intersect at right angles.

Explanation: Consider the equation of parabola, y=x2.

The slope of the tangent is provided by dydx=2x.

Consider two points on the parabola (x1,y1) and (x2,y2). given by a and b

Therefore, the slope of the tangent lines for both points would be 2x1 and 2x2 respectively.

Therefore, the equation of the tangent lines is,

yy1=2x1(xx1) and yy2=2x2(xx2)

Solve for y in this case,

yy1=2x1(xx1

64b)

To determine
Use the result of the part (a), to show that the tangent lines to the parabola x24x4y+8=0 lies on the equation of directrix.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### In Problems 51 and 52, rationalize the numerator of each fraction and simplify. 52.

Mathematical Applications for the Management, Life, and Social Sciences

#### Solve for y: 2x+3y=12

Elementary Technical Mathematics

#### limxx3x+2

Calculus: Early Transcendental Functions (MindTap Course List) 