prove-the-line-tangent-to-the-hyperbola-fracx2a2-fracy2b21-at-the-point-leftx_0-y_0right-has-the-equ

Question

Prove: The line tangent to the hyperbola
$$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$
at the point $\left(x_{0}, y_{0}\right)$ has the equation
$$\frac{x x_{0}}{a^{2}}-\frac{y y_{0}}{b^{2}}=1$$

Suman Saurav Thakur · Accepted Answer

discipline or 50. We need to prove the line attention to the hyper bola given hyper valet&#x27;s X squared by a squared minus y squared by B squared equal to one at ex not from a Why not? Is x x not by a square minus Why? Why not buy base quite equal to one? So we have to prove this. Let us suppose this to be question number one. Now we will be starting with, uh, this point. It&#x27;s not violent. Why not lies on the parabola? So the question number one becomes it&#x27;s not a square by square, minus whiner square. A B is quite equal toe one. So if you need to find the value of violent square by B squared, this will be why not a square. But be square, excellent squired by a square minus one. So we will be using this value afterwards. Now let us try and Royal off a sketch off the hyper Gola and the tangent. Okay, I probably will look like this. And if this is any point x, not comma, why not? So tangent would look like Yes, And when that&#x27;s not why it is the point of tendency So first of all, we&#x27;ll be finding the slope of pendant slope off the tangent. Okay, stop off any present at this hyper Bhola, which will be found out by differentiating the generally question. All given a question off Hyper bola with respect to X minus y squared by B squared equal toe D by D. X one. So since differentiation follows distributive law so this will be DVD X axis Squire by a Squire minus DVD X. Why squired by B squared equal to zero because constant term have a differences in equal to zero No one by a square and one baby square are constants. So this can be taken outside the i d X access square minus one by B squared DVD X Waske or equal to zero. So access choir has differentiation to works. So two x by is choir minus to Why by be square Do you ever the X equal to zero? This condition is to why, by being squired de vie ready X equal took two works by a square nor multiply both sides by being square by two y This will become Do you have any X equal to to expire a Squire and to be square by two. Why do you will get canceled out? So be squired my square. That&#x27;s my wife. So this is the slow attendant at any general point. For in a particular point we confined slow. We can find the slow attendant at It&#x27;s not common. Why not? So we&#x27;ll be plugging the value off X as X not. And whereas we&#x27;re not, this will be be a square X not by is square. Why not? Now we&#x27;re building using this value to write the question of tangent. So we appoint next month. Why? Why not? We have slope so point slow form off a straight land. Why minus why not equal toe d y by d. X That slope at It&#x27;s not Why not x minus x Not so. Why am I? So I am not Will be equal toe Be a Squire X not my Esquire. Why not X minus X? Not okay. So let us cause Multiply with Why not? Why minus why not into Why not equal toe b squared ex not X minus that&#x27;s not multiply. Both multiply or opening the back it with Why not his in distributive law? Why Why not? Minus one. The square equal Toby Squired ex Not this is square with luminess it is X minus X not not divide. Both sides were b squared. It will be Why? Why not by the square minus Why not a square by B squared equal toe X marked by a square X minus X Not now. We have already found the value of wise. Why not square the B squared? Which waas here? X marks Squire by a square minus one. So let us plug in. Okay, this is why? Why not? My b squared minus X More squared. Bye. A square minus one. Now it is open the bracket XX not by a square minus x more squired by a square. So why? Why not to square the be square minus x not square by square Bless when equal toe xx not Why is choir minus x not square by a square. So this can begin sellout and little subtract why we&#x27;re not based world from both the sides. We will be having xx marred by a square minus Why? Why not buy base square? So this can also be written as xx not my a square minus. Why? Why not by B squared equal to one, which is the required A question off the attendant. Thank you.

Suman Saurav Thakur · Answer

we have 49 number problem in which we need toe that the line tension toe the lips access square by square plus y squared by B squared equal to one. It&#x27;s has the question at point. That&#x27;s not common wine. And this is the point attendance tendency off the line. Okay, uh, is X ex not by a square, plus Why? Why not By being square equal to one. Okay, so let&#x27;s just, uh, draw a rough sketch of the lips X. Why? Okay, if this is the point of tendency, so attendant would look like this. This intendant And this point is it&#x27;s not comma. Why not now? One thing we can see that. Yes. Yeah, the point. Excellent. Why not lies on their lips Access squired by X squared plus y squared basically equal to one. So it&#x27;s not a square by square. Plus, why not Squire? By being squared equal to one. Okay. And we have a question number one as this, given the lips. Okay. Now, first of all, will have to find the slope of tangent slope off tangent. Um, an ellipse on these lips. First of all, we live finding through containment at any point that is gentle, uh, value for slow attendant. And then we&#x27;ll be plugging in next, not while not toe. Get the slope attendant. At that point, that&#x27;s not why not? Okay, so this can be found out by simply differentiating because the number one be both sides with respect to at both sides. This is access Square by a Squire. It was a white square by B squared equal toe DVD X toe. One. No, this can be written is D by D. X access choir by a Squire plus DVD x oh, I square by B squared equal to zero because differentiation off constant term is always zero now, one by a square is constant, so one by a square DVD X x x squared and one will be a square is constant as well, equal to zero. So when by a square into access where we&#x27;ll have differentiation two X plus one by B squared two way in two d y by the X equal to zero. Okay, now, if we can divide both sides by two, will be getting X by a square. Plus why, by B squared equal toe. Do you have any X this is not equal to this isn&#x27;t? Yeah, into De bye bye. The X equal to zero. Now we&#x27;ll subject expert is quite troubled. Decides why by b squared into divide body X equal to minus X Y s quest. No dividing both sides by why buy baseball? We will be getting d y by D x equal toe. Okay, so that&#x27;s really minus. That&#x27;s my Why be square by square. So this is the value off a slope attendant at any point now we have toe find the slope of Tencent at the point x Not common. Why not? So just plug in the value off X and y as it&#x27;s not common, why not so divided the accepted excellent comma. Why not will be ableto minus X Not by why not be square by a square. Now we have the slope and we have the point that&#x27;s not violent through that ancient passing, so point slope form will give why minus why not equal toe slow d y by D. X at ex not comma. Why not? And x minus x Not so. Why am I so why not equal toe minus X? Not by why not B squared by a square X minus X Not Yeah. Okay. No, Um let us cross multiply. Why? Why not? This will be why? Why not Mind Once squired equal to be squired by a Squire X x not bless ex, not squared Now from a question number two we have already found Why not square equal to or we will be Just plug in the value off Why not aspire from equation? But to question number two waas, it&#x27;s not a square by yes square plus wires. Choir will be critical to one. Okay, so why not square? But the square will be equal to one minus. It&#x27;s not a square by a square. Okay, so we&#x27;ll be utilizing this value how we can utilize. So let us divide both sides of this equation but be inspired. We will be getting Why? Why not buy be square minus y squared by B squared equal toe minds and by a square xx not my access. Excellent square. We&#x27;ll be plugging in one risk Wahba basis Very equal toe this value So why? Why not buy Be square minus over minus excellent square by square next month square by square equal tow minus one by a square. Xx not place ex not square. This will become Why? Why not buy? Be square minus one plus ex not square by square and let us open the bracket using this individual law but to minus XX not by a square minus X not squared Bye Is quest Yeah Okay. You So this is minus negative. The sign Miss, take over here. This was except not so this will be negative Jackson Square here also it will be negative. He&#x27;s also negative so it will be positive. So let us plug in negative Negative, negative and this will become positive. Okay, now if he&#x27;s a protect this term acts not a square by square Both this from both the sides This will get canceled out. So we&#x27;ll be getting Why? Why not by B squared minus one equal to negative off XX not by a square. So let us add exacts not by square to both the sires We&#x27;ll be having x x nord by a square place Why? Why not my be square equal to one? Because when we look at paying to be positive on the agency this is the required equation hence proved Thank you

John Peter · Answer

the equation off the line is y equals K X plus. In the equation. Off the ellipse is X squared over a squared plus y squared over B squared equal one so often multiplication or the equation boy a square. The square We have X squared B squared, plus five square square equal is square. The square No X squared the square, plus oh key X plus in all square is where. Equal a square the square so it&#x27;s a square piece square, plus K square X square a square plus two K x in a square, plus in a square a square, equal a square square, then take the square plus K square a square. Common factor. Boy X square plus two key in the square X plus in the square square minus square. The square equals zero. Since is square E X square plus BX plus C equals zero is obtained, then the equal P square minus four. Easy now d equal four K square in a square. A power four yeah minus four, multiplied by P square plus K square square and in a square, a square minus a square. The square. This will be equal to 40 square and square A power four minus four p square in a square a square right plus four a square piece square, minus four K square in a square. A power four lost. Four key square. Evil food P square. After simplification, we have the equal four. A square, a square multiply pie key square a square loss. He square minus in the square. Yeah, since four. A square square greater than zero. So K square a square plus B squared minus in square. Greater than zero. The line cuts the ellipse in two points. End que square a square plus B squared minus in a square, less than zero. The lion Do not cut the Alps, and case square X squared plus B squared once in square will be equal to zero. The lion cuts the lips in one point. So the equation off the line Tension toe the ellipse at the point. X note on why note would be equal to X X mode over a square. Plus why? Why not Overpay Square equal one? Yeah, yeah,

Sriram Soundarrajan · Answer

Hello. Good. If you&#x27;re going to sort problem Number 74 here. We had to find the equation off the panda, which is passing through the Hi, Barbara. It is just a big square by square minus lightsquared by the square. You could do so, uh so let this camera agnostic square The square minus life square is got Chicago here. Square into the square. Medication off that light and be like kids plus 10 south shooting Biker The cakes percent your cake It&#x27;s a squared B squared minus cake&#x27;s plus And those square into your skirt. This is a Goto. Yes. Quit. It&#x27;ll be square for what we had to do. It means it squarely. Square minus que square Every square is square My enough do kids in a square minus in square ISC wear a good look He&#x27;s squaring Toby Square full beef choir minus que square in tow square and a few square minus. Don&#x27;t get in Indo isc Latics going us in, squared into his square minus is square in the b squared equals zero So yeah, Marti&#x27;s This is Yeah, this is B andi, is it? See? So we have proof. I&#x27;m easy for group B squared minus four a c. So which is given by this one is four k square in square. Here is to food plus my enough minor for raising right foreign do Is the square minus K square? Yes. Square in the seas minus in square is square minus years square. Be square So do you regret getting it&#x27;s legs for que square and square into your toe The bar for my ah bless for beef square and square square plus four Yes, square because apart for minor, full case square and square it with a powerful my full case choir paper about for into the square I was using cargo for its square beef in the minus case square Esquire plus beef in squared before is quieting beef. One is greater than zero or this I know if did. Hey, Esquire minus B squared minus in square. If less than zero, it guard the hyper Ebola. You got the high bar ball up in the room points. Okay. If okay, give concerning your case. Good case square a square minus B squared, minus and squat is get a dank little Do not cut the hyper Ebola. It is not good did. Hi. But born up if case car is minus B squared minus in Quincy Carter zero. It can&#x27;t be hyper bola. I did. Hi. But, Bora, I don&#x27;t find one point. An indication understanding can be There are like Extender X Nordby, If glad minus my window. I not by the square is a good line. Thank you.

Doruk Isik · Answer

in this problem, we&#x27;re going in question. And where s a final question of tension line that important? It&#x27;s gonna Why not be known at the question of pencil and move you or y minus order Physical, too slow or David or dysfunction? It will get given point off if I explain it&#x27;s Exner, for we first need to pack plate conservative industry point. So let&#x27;s differentiate this given in question. We have to ex or a sprayed minus two r or B squared y prime Izzy zero from this. Then we see that what prime is equal to thanks over a spread times bees for a while or redirect this one as a virtually dissect given point. We would find why five to be equal to bees. Word over a scrape month like ex. Not we&#x27;re going up that if that is, why primarily given. Let&#x27;s write. Be questioned tangling again. We have one minus one is equal to be scared over a squared times X No, no. Why not? Well, like my ex months ex, not now. Let&#x27;s multiply both sides by a straight times. Why not? Um, you would end up with a straight times wine up on four minus a spring times. Why not? Squared is equal to B squared X nine times, X minus B squared pig snouts crab we can write for do this One is a spurred by not time for minus be spared. Excellent X that is equal to be screwed. That is he going to piece Fred Ex not spread minus a spring. Why not? Square? All right. Um well, that should be the underwear up this b a screwed. Why not? Spread minus be spurred. Excellent spur or now it be. Divide both sides. Fine. A squared times be spurred. You would end up with eggs. Not Or why not? Times why divided a B squared minus extra funds exit the life by a spread is equal to why no, it&#x27;s great. Be spread minus x Not spread you on a squared or this is Samos Exact times like so race bird minus one on time for lawyer will be spread is a question Excellent Squared away. Scrape plus or minus one up spread over these four. So basically I most vital signs are negative. Or now look at the right inside here and look at the origin of the question. Is he blinking point? Excellent order into the ordinary question. It would end up with one. And since right that side is the same, we&#x27;re going to say that frightening side, is it? Well, so then the question of potential lead is excellent. Point X provide a sprig minus one for a while, but he&#x27;s very busy.

Linh Vu · Answer

a question X squared over a squared minus y squared over p square. You could you one. And then we need to find a dent in line and a point x zero y zero here. Recalled in a tent in line by formula. We have the y minus wise. There are and we go, June. Uh, why Graham Times x minus X zero. And we need to find a white Bram in this case here, just in the embassy. The Riverdale. So if we don&#x27;t invest in the river to hear began that you x number a square minus June wine times y prime about the p square coaches zero from here. Gonna when prime with Iko Do No. Ah, too concerned with juice. So we get and will be b squared over a square. I, um Why? And now we put this one into this formula, and then we get the why minus guys are echo to be square over a square. X is an excellent Suazo now, because we need to look in the front and then x minus X zero Now do didn&#x27;t do in the cross modification. We have, uh, as a wise oh, why minus is Owais. Oh, square echo to the B square, extra X and then minus Be square X zero square moving the term around. We have eyes square. We have the be square X zero. Thanks on da We will have This would be, Yeah. A B squared x zero x minus I square wise or why they were ICO Judah be square X zero square minus y squared wise or square. Now dividing everything. My dis quantity. Michigan the X zero X Ah, but the a square minus Wiesel. Why over the peace? Quiet! You go to one and this would be the question on the front line.

Problem

Use the results in Exercises 49 and 50 to show th…

Problem 50 Hard Difficulty

Prove: The line tangent to the hyperbola
$$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$
at the point $\left(x_{0}, y_{0}\right)$ has the equation
$$\frac{x x_{0}}{a^{2}}-\frac{y y_{0}}{b^{2}}=1$$

Answer

$\frac{\pi_{0} x}{a^{2}}-\frac{y_{0} y}{b^{2}}=1$

Related Courses

Calculus Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Heather Z.

Caleb E.

Michael J.

Joseph L.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 10

Video Transcript

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals

Heather Z.

Caleb E.

Michael J.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$\frac{\pi_{0} x}{a^{2}}-\frac{y_{0} y}{b^{2}}=1$

Calculus Early Transcendentals

Heather Z.

Caleb E.

Michael J.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Heather Z.

Caleb E.

Michael J.

Joseph L.

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals