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
Show that the ellipse $ x^2/a^2 + y^2/b^2 = 1 $ and the hyperbola $ x^2/A^2 - y^2/B^2 = 1 $ are orthogonal trajectories if $ A^2 < a^2 $ and $ a^2 - b^2 = A^2 + B^2 $ (so the ellipse and hyperbola have the same foci)
Video Answer
Solved by verified expert
This problem has been solved!
Try Numerade free for 7 days
Like
Report
Official textbook answer
Video by Suman Saurav Thakur
Numerade Educator
This textbook answer is only visible when subscribed! Please subscribe to view the answer
01:10
Frank Lin
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 5
Implicit Differentiation
Derivatives
Differentiation
Askjfas S.
November 18, 2017
32t
asf
xjc
Harvey Mudd College
University of Michigan - Ann Arbor
Boston College
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.
0:00
Show that the ellipse $ x^…
01:03
Show that the ellipse $x^{…
03:27
04:05
If a point $\mathrm{P}$ on…
we have a question we need to show that access choir by a Squire plus Y squared but being square equal to one. And the hyper bola. This is the lips and hyper bola access squired by a square minus y squared but be square. Well to one article. Electronic trees are total project remain means if a find slope of this divide by dx and slope of this let us right slope of tended on this which is uh if her lips and D Y by dx slope of hyper Ebola so productive they should be minus one. This is the meaning of all to the military secretary over here. So let us find dy dx differentiate with respect to X two x by a square plus two. Y. Do I buy DX Bible Square Equal to 0? Which means divided by D X will be equal to minus two X by a Squire into being squired by to Y. Two and two will get cancelled out. So do you have any excess hm minus X. B squired by esquire white. This is do you have any exit E. For ellipse? Similarly for help. Ebola. If we differentiate will be getting two X by a square -2 Whereby Be a Squire. Diva Body x equal to zero. Okay so uh D Y by D X will be equal to two x by a Squire into b squared by do it. These two will get cancelled out. So be esquire X by uh this is a Squire. Why this is D Y by the X slope of tangent at hyper bola. If we do like this we'd be getting uh minus X. Being squired by is quite why multiplied by be a square X by is quite a way. They should be equal to minus of one. Okay. Which means this is quite a way which means uh B squared B squared minus X squared equal to minus is square is square. Well y squared some mines mines will get cancelled out. So we have is it be as quiet right divided by a square is quite equal to Why? Square by Expert Equation # one. Now from any question let us get the value of access required by Y Squire. If he uh could just get the values. Mhm Okay, is everything good? Yes. Yeah. Let us get values from any one of these. So let us uh to get the values, we will be subtracting if this is a question number one, the same question number two. For the sake of simplicity will subtract any question number one from the question number two. The art form will be doing that technique. Question # two from the question # one. So this will be access choir by a square plus. We're squired by being square equal to one and minus. Access square by square. Mine's white square but it is quite equal to one minus plus and minus. So one will become right inside it will become zero and access square will be taken as common. So one by a square -1. But capital is squared bless. Why square will be taken as common? One by B squared minus one by capital B square equal to zero. Yeah. Uh so this is extra square and a square minuses square by square is what equal to Y squared one by b squared minus one by small, B squared Y squared more B squared minus capital B square being square. So we need to find the value of this. X squared, Y Y squared by X squared from here was squired by esquire will be able to is squared squared. Okay, we are dividing both sides by access. Quite so yeah, and this will be equal to be a squared B squared by a square square into is Squire minus a squared by small B squared minus capital base question Okay now we'll be plugging in the value in equation number let us say question number three in question number three. So we'll be getting could be a squared B squared by a square. A square equal to this being squared, B squared by Smalley square. Capital is squared square minus a squared by the square minus business. So these people get canceled out Here. It will be one. So the cross multiplying will be getting capital a square mind. Smalley square equal to a small base square minds Capital base square. Okay, so if we add capital D squared to both the sides, A squared plus b squared and again smaller square to both the sides. So this will become a squared minus B squared. So this is bro, thank you.
View More Answers From This Book
Find Another Textbook
Oregon State University
Idaho State University
01:48
Question 3ptsThe math department faculty at large university wanted…
04:02
8. The null and alternative hypothesesFor each pair of hypotheses that f…
01:46
Complete the square of the given quadratic expression_ Then, graph the funct…
03:18
Plutonium-238 is rdioactive element that decays over time into less harmful …
02:43
Find the exact length of the curve x=tsint 0<t<l y=tcostx(t)-tsin(…
04:26
Consider triangle ABC like the one below: Suppose that a = 44, c=47_ and A-3…
Question 1: Duality [30 points]The following exercise walks you through …
06:03
Exercise 1Assume that the time spent online by customers is normally dis…
02:57
At 3 PM a red car is 90 miles north of Bellevue and a blue car is 100 miles …