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Consider the ellipse$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$where half the length of the major axis is $a$ , and the foci are $( \pm c, 0)$ such that $c^{2}=a^{2}-b^{2} .$ Let $L$ be the vertical line $x=a^{2} / c$(a) Prove that L is a directrix for the ellipse. [Hint: Prove that PF/PD is the constant c/a, where P is a point on the ellipse, and D is the point on L such that PD is perpendicular to L.](b) Prove that the eccentricity is $e=c / a .$(c) Prove that the distance from $F$ to $L$ is ale $-e a$

a. $\frac{P F}{P D}=\frac{c}{a}$$L$ is a directrixb. $\frac{c}{a}=e$c. $\frac{a}{e}-a e$

Calculus 2 / BC

Algebra

Chapter 8

Analytic Geometry in Two and Three Dimensions

Section 5

Polar Equations of Conics

Polar Coordinates

Introduction to Conic Section

Oregon State University

McMaster University

University of Michigan - Ann Arbor

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So we're giving the following information about ah, on the lips This ellipse described by the equation X squared, divided by a squared plus y squared divided by the square A circle to one and we want to prove that the line xs equal to a square divided by sea is the directories for this ellipse to do so we're gonna prove that the line segment P f which is this one right here, this one degraded by the line segment p d Just just this one is equal to the constant See over a to start off. We're gonna take the following equation for an ellipse in polar question Polar Coordinates, which is art. A sequel to E. Tim's game divided by one plus E Co Cynthia. And for an angle of fate icicle to zero, we'll have the following a question. Our physical to eatem ski. You know I want plus e since the coastline of today's just Europe and that is equal to a minus c. So remember that the distance from the center to this Burke takes over here is a the distance from the centre toe. One office folks, I it's C and so the radius in this case, just gonna be the difference between A and C, which is this segment right here. This is our So now we're just gonna sold for? We're gonna take this equation and sold for dis constant right here. We'll have e tem ski if you go to a minus C times one plus e And so we also know that the constant e sequel to see Divided by a So now I'm Minister working These new networks will have eight times. Kate is equal to a minus c tim's one plus e well, to know that he is equal to see divided by a So now we'll have the line segment pf just this one right here. P f That's it for this line segment we have that the distance, the existence from the faux say to the point is just gonna be our co Cynthia. What? And so for this point right here, we'll have that our radius over here just gonna be e temps que you ready for that one plus e co Cynthia Well, we have a stated that eatem ski. It's just equal to e a minus C attempts one plus e. He's just gonna beat the same. The denominators stays the same one plus eight times goes in theater. Well, we also have stated that he's see divided by a we'll have a minus C plus one plus c divided by a You better buy one plus see temps, eh? Cosign theater. And so now I'm gonna start working with gel giraffe over here. So we'll have this same line segment, which is just gonna be a minus. C. I'm gonna distribute this term right here. I'm gonna multiply. Actually, I'm gonna multiply everything through so we'll have a temps one just want a plus c divided by eight times eight is just see so negative C Times one is just negative c and so negative. See, divided by a It's just negative. C squared, divided by a So now the denominator stays. I'm gonna multiply everything through buying a over here, someone a month supply everything might a on the top and on the bottom we'll have one plus c divided by eight coast and failure. So we'll have this age was gonna become a square. So these two see terms cancel each other minus C square. Since I have multiplied by in a term. So over here we'll have a plus C co, Cynthia. And so now that we have this term break here, we have our we have a relationship or be square for our inner lips, which is B squared is equal to eight square minus C square. And this is exactly what we have over here. This is the same. So we can say that this term right here it's gonna be be square divided by a ce temps cosign eight plus c times coasting theater. So we're gonna step right here and we have the chairman on a question for the line segment pf now we're gonna work our way through the segment slain segment P. D. So, in this case, P d just gonna be the distance between the X coordinate, which is X is equal to a square. Their embassy in our location for point D So the ex according it for a line Sigman Just gonna be a square. You ready, Casey? In our point, it's just located from this from the origin. It's located at a distance. See? Sorry C plus our cosa fada. So we have our distance from the center which is our point of reference. This is gonna be this sea part. Plus are our CO Cynthia, which depends on the angle at the point makes would respect to the folks I And so if we distribute this native sign will have a square divided by C minus C minus arco. Cynthia. And so this term we're just going to say that we're just gonna multiply everything through because we already have the chairman who have discs gonna be equal to a square minus C square. Yeah, but if I see I'm just I'm just multiplying this seat term times this one minus So we have determined that our in this case, which is the point, which is this point right here it's also equal to our It's just be square derided by a plus C Tim Scottson theater times that CO Cynthia. So let me rewrite that. So the point The lines England Petey just gonna be a square minus c square. Where did they see which is just the multiplication of distorted the addition of this too minus be square but by a plus. See temps go Cynthia times on extra co cynthia to remember that this term right here is just our And we determined that one right here. So now we have determined that we're just gonna at all of these terms. I want to multiply our coast and fate over here. And remember that we said that B square is equal to a square of minus C square, so we'll have b squared, divided by sea minus B squared co Cynthia, you better buy a policy Cosa stato. And so this is gonna be able to monitor right over here. So this term is just gonna be equal to be square. I'm just gonna multiply everything on the bottom. Sort of common denominator is just gonna be C attempts a plus. Seiko Cynthia Dempsey. So we're gonna multiply everything r B terms is gonna multiply by this and this term is gonna multiply. Break this one. So be squares times a plus. See temps coast and theater minus c Tim's be square cosign fado. So I'm gonna I'm gonna multiply everything through over here and I'm gonna distribute this. See? So we'll have B square times, eh? Plus B square. Come see Tim Scottson theater minus C tends to be square times. Cosine theta he read it by 8 a.m. c plus C square, coast and freedom. And now we're here. We can see the district terms cancel each other. So finally, I'm going to rewrite this. A speech. Square terms, eh? Nope. Sorry. Be square temps, eh? Minus. Oh, Sorry. Divided by eight m c plus c square cosa stato. So we have determined this over here. It's a nice time that we take the ratio of the two of them. So we have the ratio of P F. Goodbye. P d must be equal to see the red light, Eh? So we're gonna work with the term So be earth PF Sorry. She's just one right here. It's just gonna be able to be square. We'll have b squared divided by eight plus seat temps Coast in a theater divided by be square temps, eh? Times see tempts a plus C plus go, Cynthia. So you can see right here that I just pulled one c term over here. One of her common sees And their reason to do that is because we'll have B square, Tim, see times this term right here. A plus C plus co Santa divided by the square tem See times a plus See temps coast in theater So we can see that this to cancel each other and this just becomes one. So this term it's just one. So they will have be square Tim seat divided by the square temp See, in these two A's Sorry, this to be square are also one Oh, just any sorry. So finally we have demonstrated that the ratio of p f divided by P. D. Using our directories s X equals a squared. Where did I see he's actually our directories Since our ratio of p f divided by P. D. Is equal to see the brighter by a So now on part B, we want to prove that the eccentricity so hee is equal to see the bite of a And so he is defined by the ratio p f divided by P. D. To the ratio of these two distances the ratio of the distance between the point off, any point in the lips to its foresight to the ratio of the point of the distance off the distance between the point and the directories. Hope sorry. And so in this case is ratio. We have proof over here that is sequel to see Divided by A with our directories. So this is equal to see divided by a So finally, when to prove that this line segment of the foresight to the it's directories, which is represented by a LSO this is L It's equal to a well by e minus. He divided by a said to do so we're gonna take the line segment f l. So it's false eye over here, which has co ordinate See, So remember that our folks I are located are plus or minus C coma zero. So in this case, it would take the postive. Suppose I she's gonna get this one right here. You don't have a just a coordinate of C. It is a distance See from the center sword directories is this equal to X is equal to a square. Even if I see. So with this information in mind, we're gonna take a square, right? I see minus the distance, see from the origin. So distance right here is just a squared divided by sea. And so now we know that e the sequel to see Divided by a So in this case, We can also say that C e sequitur E times A and we can substitute that over here we'll have a square divided by E temps, eh? Minus attempts A and so one of this ace cancel with the square. So we'll have a united by e minus. E temps, eh? And in this case, we have proved that the line segment from the foresight to this directories is equal to a divided by E minus three times a. So this problem we have proved that the directories has co ordinates a squared divided by sea by proving that the line that the ratio of the line segments PG rated by P Sorry, pfc, But LAPD, it's equal to see divided by a We also proved that e physical to see divided by eight by its definition which which is the ratio off the line segment between the point to folks. I focus in this case and point to direct tress which was just the result offer proof over here, which is just see divided by eight. By using this by using all Durant and the definition offer an ellipse. And finally we have proved that the lines that the distance between the focus and the direct tress is equal to a divided by E minus times, eh? By using the definition off its location in the X axis.

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