Explanation Given information: Given that a > b > 0 and c = a 2 − b 2 , P ( x , y ) is a point on the ellipse ( x a ) 2 + ( y b ) 2 = 1 . Also F = ( c , 0 ) , e = c a and vertical directrix D at x = a e . Proof: The distance between the point P and F is given by: P F = ( x − c ) 2 + ( y − 0 ) 2 P F = ( x − c ) 2 + y 2 The distance between the point P and the directrix D is given by: P D = | x − a e | Consider the equation P F = e P D . Plugging the values to check if the equation holds true: ( x − c ) 2 + y 2 = e | x − a e | ( ( x − c ) 2 + y 2 ) 2 = ( e | x − a e | ) 2 ( x − c ) 2 + y 2 = e 2 ( x − a e ) 2 ( x − c ) 2 − y 2 = ( c a ) 2 ( x − a ( c a ) ) 2 ( x − c ) 2 + y 2 = c 2 a 2 ( x − a 2 c ) 2

4th Edition

ISBN: 9781319050740

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 11.5, Problem 65E

To determine

To prove that a point P=(x,y) on the given ellipse satisfies PF=ePD for the given conditions.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.5, Problem 64E

Chapter 11.5, Problem 66E

Students have asked these similar questions

Question Is the function f(x) shown in the graph below continuous at x = -5? f(z) 7 6 5 4 2 1 0 -10 -6 -5 -4 1 0 2 3 5 7 10 -1 -2 -3 -4 -5 Select the correct answer below: The function f(x) is continuous. The right limit exists. Therefore, the function is continuous. The left limit exists. Therefore, the function is continuous. The function f(x) is discontinuous. We cannot tell if the function is continuous or discontinuous.

The graph of f(x) is given below. Select all of the true statements about the continuity of f(x) at x = -1. 654 -2- -7-6-5-4- 2-1 1 2 5 6 7 02. Select all that apply: ☐ f(x) is not continuous at x = -1 because f(-1) is not defined. ☐ f(x) is not continuous at x = −1 because lim f(x) does not exist. x-1 ☐ f(x) is not continuous at x = −1 because lim ƒ(x) ‡ ƒ(−1). ☐ f(x) is continuous at x = -1 J-←台

Let h(x, y, z) = — In (x) — z y7-4z - y4 + 3x²z — e²xy ln(z) + 10y²z. (a) Holding all other variables constant, take the partial derivative of h(x, y, z) with respect to x, 2 h(x, y, z). მ (b) Holding all other variables constant, take the partial derivative of h(x, y, z) with respect to y, 2 h(x, y, z).

Chapter 11 Solutions

Calculus: Early Transcendentals

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 6PQ Ch. 11.1 - Prob. 7PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To prove that a point P = ( x , y ) on the given ellipse satisfies P F = e P D for the given conditions.

Solution Summary: The author proves that a point P=(x,y) satisfies PF=ePD for the given conditions.

Concept explainers

Videos

To prove that a point P=(x,y) on the given ellipse satisfies PF=ePD for the given conditions.

Want to see the full answer?

Chapter 11 Solutions