Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

A tangent line is drawn to the hyperbola $ xy = c $ at a point $ P. $(a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is $ P. $(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where $ P $ is located on the hyperbola.

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Clarissa Noh

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

02:12

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Derivatives

Differentiation

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

04:16

A tangent line is drawn to…

0:00

06:01

Yeah, it's clear someone named Reed here. So we have. Why is equal to see next to the negative one power. We can rewrite it as follows so the derivative, when we find it, you get negative C x to the negative to power the tangent line Therapy of X one comma See over X one is why minus c over X one over X minus X one is equal to negative C next one the negative to power and this becomes why is equal to negative c x of one the negative to power X plus two c ex of one, the negative one power. So the Y intercept iss to see over at someone that iss the Y coordinates of P Um, that's two times the white cornet and you got the X intercept. We make zero equal negative C except one too negative to power. Thanks plus two C texts of one to the negative one power, and that gives us to accept one. And that's twice the X coordinates of P. So P is the midpoint off the line segment formed between the intercept for part B. We got that Why is equal to see X to the negative one. So the derivative is equal to negative. C x, the negative to power we want The tangent line simplifies to why is equal to negatives CX of 12 negative to Power X plus two c ex someone the negative one power. So the why intercept is to see over X someone when we get the X intercept by making it zero and we've bought two x someone we know that the area of the triangle is equal to have of the X intercept times the Y intercept, which is to see divided by ex someone on two ex of one is equal to to see and that's independent of X sub one so p.

View More Answers From This Book

Find Another Textbook

01:47

Let f(z) There are two lines with positive slope that are tangent to the par…

02:25

Let f(t) = (t2 + 2t +5) (3t2 + 5) . Find f' (t). f'(t) =Find f…

02:28

Question 63 ptsThe following data were collected by counting the num…

00:51

Write an equation for the function graphed aboveyCheck Answer

02:05

The answer above is NOT correct:point)Find the coordinates of x in t…

02:10

Consider a square city that is 100 km?. Suppose you design cellular system f…

01:53

Transform each graph as specified below:(a) The graph of y=f (x) is show…

04:39

Activity 1: Evaluate the following trigonometric equations. (5 points) 1 Cos…

02:34

Certaln radionctive maferial decays such way (hat the Mass remalning alter V…

01:34

Use the Factor Theorem to determine whether x -2 is a factor of P(x)=x"…