/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 77 Show that an equation of the lin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 77

Show that an equation of the line tangent to the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) at the point \(\left(x_{0}, y_{0}\right)\) is $$ \frac{x x_{0}}{a^{2}}+\frac{y y_{0}}{b^{2}}=1 $$

Short Answer

Expert verified
Question: Show that the given equation is tangent to the ellipse at point \((x_0, y_0)\). Given Equation: \(\frac{x x_0}{a^2} + \frac{y y_0}{b^2} = 1\) Ellipse Equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) Answer: We have shown that the derived tangent line equation, \(\frac{x_0 x}{a^2} + \frac{y_0 y}{b^2} = 1\), matches the given equation. Therefore, the given equation is tangent to the ellipse at point \((x_0, y_0)\).

Step by step solution

01

Find the general equation of the ellipse

The given ellipse has the general equation: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$
02

Determine the slope of ellipse at \((x_0, y_0)\)

Now we need to find the slope of the ellipse at point \((x_0, y_0)\). To do this, we will first implicitly differentiate the ellipse equation with respect to \(x\): $$ \frac{2x}{a^2} + \frac{2y}{b^2}\frac{dy}{dx} = 0 $$ Now solve for \(\frac{dy}{dx}\) to find the slope: $$ \frac{dy}{dx} = -\frac{2x}{2y}\frac{b^2}{a^2} $$ Now, we can plug in \((x_0, y_0)\) to get the slope at that specific point: $$ m = -\frac{2x_0}{2y_0}\frac{b^2}{a^2} $$
03

Obtain the equation of the tangent line to ellipse

Using the point-slope form, we can write the tangent line equation at \((x_0, y_0)\). The point-slope form is \(y - y_1 = m(x - x_1)\). $$ y - y_0 = m(x - x_0) $$ Where \(m\) is the slope we calculated in Step 2. Plug \(m\) into the equation and simplify: $$ y - y_0 = -\frac{2x_0}{2y_0}\frac{b^2}{a^2}(x - x_0) $$ Rewrite the equation and simplify a bit more: $$ \frac{b^2}{a^2}\left(\frac{x_0(x - x_0)}{y_0}\right) + y = y_0 $$ Rearrange the terms: $$ \frac{x_0 x}{a^2} + \frac{y_0 y}{b^2} = 1 $$
04

Show that our derived tangent equation matches the given equation

The tangent line equation we derived is: $$ \frac{x_0 x}{a^2} + \frac{y_0 y}{b^2} = 1 $$ Which matches the given equation, thus proving that the equation of the line tangent to the ellipse at the point \((x_0, y_0)\) is $$ \frac{x x_0}{a^2} + \frac{y y_0}{b^2} = 1 $$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Implicit Differentiation
Implicit differentiation is a technique used in calculus to find the derivative of a function without explicitly solving it for one variable. When dealing with equations like the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), both x and y are interrelated, making it impossible to isolate y and differentiate directly.

The process begins by differentiating both sides of the equation with respect to x, treating y as an implicit function of x. For instance, when we differentiate \(y^2\), we apply the chain rule: \(2y \frac{dy}{dx}\). The derivative \(\frac{dy}{dx}\) represents the slope of the tangent line at a given point on the ellipse, and through this method, we can compute it without the explicit expression for y.
Slope of Ellipse
The slope of an ellipse at any point is essential for understanding the direction in which the ellipse is moving at that point. It's particularly useful when you want to find the equation of a tangent line at a specific point \(\left(x_0, y_0\right)\) on the ellipse.

By using implicit differentiation, the slope \(m\) at a point on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is given by \(m = -\frac{x_0}{y_0}\frac{b^2}{a^2}\). This negative reciprocal relationship between \(x_0\) and \(y_0\) reflects how the tangent line is perpendicular to the radius of the ellipse at that point, which makes sense since tangents are always perpendicular to radii at the point of tangency.
Point-Slope Form of a Line
The point-slope form of a line is a straightforward way to write the equation of a line when you have a point and the slope. This form is written as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \(\left(x_1, y_1\right)\) is a point on the line.

For tangent lines to an ellipse, once we find the slope \(m\) at a particular point \(\left(x_0, y_0\right)\), we can substitute these values into the point-slope form. Then, algebraic manipulations allow us to rearrange the equation into a more familiar linear equation format. It's a useful formula because it directly uses the properties of the point and slope without requiring any additional transformations.
Equation of an Ellipse
The equation of an ellipse represents the set of all points \(\left(x, y\right)\) that form an elliptical shape when plotted on a graph. The standard form of an ellipse equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An idealized model of the path of a moon (relative to the Sun) moving with constant speed in a circular orbit around a planet, where the planet in turn revolves around the Sun, is given by the parametric equations $$x(\theta)=a \cos \theta+\cos n \theta, y(\theta)=a \sin \theta+\sin n \theta.$$ The distance from the moon to the planet is taken to be \(1,\) the distance from the planet to the Sun is \(a,\) and \(n\) is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of \(n\) produce loops for a fixed value of \(a\) a. \(a=4, n=3\) b. \(a=4, n=4 \) c. \(a=4, n=5\)

Consider the parametric equations $$ x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t $$ where \(a, b, c,\) and \(d\) are real numbers. a. Show that (apart from a set of special cases) the equations describe an ellipse of the form \(A x^{2}+B x y+C y^{2}=K,\) where \(A, B, C,\) and \(K\) are constants. b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the \(x\) - and \(y\) -axes provided \(a b+c d=0\) c. Show that the equations describe a circle provided \(a b+c d=0\) and \(c^{2}+d^{2}=a^{2}+b^{2} \neq 0\)

Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at the origin with major and minor axes of lengths 12 and \(2,\) on the \(x\) - and \(y\) -axes, respectively, generated clockwise

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The hyperbola \(x^{2} / 4-y^{2} / 9=1\) has no \(y\) -intercepts. b. On every ellipse, there are exactly two points at which the curve has slope \(s,\) where \(s\) is any real number. c. Given the directrices and foci of a standard hyperbola, it is possible to find its vertices, eccentricity, and asymptotes. d. The point on a parabola closest to the focus is the vertex.

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of the parabola \(y^{2}=4 p x\) or \(x^{2}=4 p y\) is \(4|p|\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.