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Problem 51 Hard Difficulty

(a) Graph the curve with parametric equations
$$ x = \frac{27}{26} \sin 8t - \frac{8}{39} \sin 18t $$
$$ y = -\frac{27}{26} \cos 8t + \frac{8}{39} \cos 18t $$
$$ z = \frac{144}{65} \sin 5t $$
(b) Show that the curve lies on the hyperboloid of one sheet
$ 144x^2 + 144y^2 - 25z^2 = 100 $.

Answer

A.
B.Plug the parametric equations into the given hyperboloid of one sheet equation
and manipulate so that the side you plugged into reveals itself to be $100 .$

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Video Transcript

in this problem. Want to graft the curve with parametric equations. X equal 27/26 times sign of 80 -8/39 sign of 18 T. Y equals negative 27/26 co sign affects eight T Plus 8/39 co sign of agency. And Z equal 144 over 65. sign of five T. Or a parameter T. In the real numbers. Any party. We show that the curve lies on the hyper Beloit of one sheet 144 X square plus 144. White square -25. Z sq Equal 100. So in part a we are we have sketched a graph of the function is in a computer program in this case metal up we have found is Lying this curve in R. three for a parameter T. Between negative pi over 8.3 and pi over eight 0.3. Mhm. So in this case we see that the craft start at this point and it uh goes up this way. And if you look at the graph from above we will see it on the plane plane. Xy we have a sort of circle but not an exact circle. Yeah because of the equations of X. Y. And then the value of C. Is Increasing when sign of five T. is increasing after that. It is possible that the curve decrease again because we will have decreasing value of sine of T. And so on. So This sketch of the graph that in three D. He's not not that is to see But this case can imagine this care of is starting at a certain level. And then he's getting up up this hill making a sort of around in the xy plane and then moving up on the seaplane. Yeah. So this is the sketch of the graph and now barbie, we're going to calculate all this expression here on the left And we will simplify all that and we will we will get at the end of 100 with that. We have proved that the careful eyes on the hyperbole Lloyd of one sheet. So let's see that started by 144 times because we have 1 to 44 in both this term and this one we're going to put it as a common factor for tambien. Then we have that factor times X squared plus y square. So we're gonna square these two expression here. So we get this case. Mhm. But scores the first term 27/26 square times sine square of 80. This is the square of this term here. Mhm. And -2 times the part of the two terms. So minus two times 27 Over 26 times eight Over 39 times the product of the sign of 80 times the sine of agent E. And now plus the square of the second term. 8/39 square times sine square of agency. So this is the first term. But I'm going to do the other one. So this is not close yet. That all these expression from here to here represents eggs square. Now we're gonna do y square. So we have plus negative 27/26 square go science squared of eight T. Then we have minus because we have these terms here with different signs. So it will end up being negative two times the product of the two terms At least 27/26 Times eight or 39 times goes on of eight T. Times go sign of agency. Now plus the square of the second term. That is 8/39 square Times Co Sign Square of 18 T. And these represents all these is why square and now all that plus or minus. Because we have in mind is here to see that's it. That is this minus year. So we get minus continue here 25 Times C sq is 140 for over 65 square times Sine square of five T. Okay. Means that we are here calculating all these expressions is the left hand side of this equation over here. So let's put now uh, this is Z square. So what we are calculating here is 144 x square plus. 144 way square minus 25. Yes, 25 c square. And we will continue here equal 144 times. And now we can mm hmm. Combined this term with this one Because we have a common factor 27/26 square And it will be a common factor of science square of 80 plus. Co signers square of 80. And we know that that is equal to one because we have the same angle 80 and the sum of the square sign and co signed for the same angle is equal to one. So We will get here 27/26 square. That is grouping this or combining his term with the swan. No, we did the same for the last two terms and for the middle terms here and here. Wilder The Middle Times 1st and I can see that we have a common factor 27/26 times 8/39. I'm gonna less the two inside after. We take a common factor out and the negative time will be taking out too. So we get negative 27/26 times eight Over 39 times. Okay, so we get to He's factor too well, let's hear inside sign a of 80 times sine of 18 T plus because we have taken the common factor, negative factor out Plus Coz I two times factor to here with this inside two time. Uh goes on Pg Times cold sign of eight and T. And we close zebra. The curly braces here because we had this some as multiply by this. Fresh. Yeah. Now we have blasts and we do the same for this two terms, this one here and this one here and again we have the common factor 8/39 square in that multiplied by some of Science square of 18 T. And go science square of 18 T. And we use the automatic identity here again and we have one inside the branches. So we get finally, Is the term simplified to 8/39 square. Gosh, this Her bracket here and then we have the last term which would be this 1 -1 25 times 144 or 65 sq sign square of five T. Now here we are going to use uh well non formulas for from two economic tree because we're going to simplify these two terms, We know that the co sign of the sum of two angles is equal to cool, sign of the first angle times go sign of the second, one minus sign of the first angle, times Sign of the 2nd 1. For the difference of the angles we have a change of sign just right here. So with this we can conclude that okay, if we adopt is to the left hand side and right hand side of these equations we get co sign of alpha plus beta Blasco sign of alpha minus vera. Okay, what we get here we get are adding the two expressions, the terms with different signs will cancel out. So we get two times go sign of alpha time scope, sign of bitter and if we do it's this direction, co sign of alpha, minor speed minus co sign of alpha blockers, Beta we'll get mm to sign of alpha sign of bid. So we have used these two expressions here to find these quality is here and that's what we want because we had on the right side of both equations we have the products of sine of angles and the parents of sine of angles which we have in this expression So if we continue here is equal to we continue here is equal to 144 times 27 or 26 square we're going to put beside this term, we're going to put this constant term here plus 8/39 square and now we have negative or minus 27, times eight over 39 and inside the curly braces we have to sign of 80 times and 18 to is his formula here, second one And we have the factor to that's what we left factor two in each case To accurately the left side of the equation and alpha is equal in this case to 80 Mbira is agency. So we'll get that this first fresh expression is go sign of 80 eight T -18 T plus my necessary because we have a minus here co sign of something that is eight T plus 18 t and now we convert the other expression that is to Co sign of 80 plus times. Come sign of 18 t. You since his first equation here. And so we get plus co sign of 80 plus 18 T. Plus plus co sign of a T minus agency. And we close the converse is because we are done here and before that we have we had the custom sir we put it here so we closing square bracket And -25 144 over 65 sq science square of hefty. Now we can see that some terms cancel out here And inside the killer braces we have uh these two terms our sex the same with opposite signs. So they cancel out and these two terms this one and this one are the same. And so this is equal to 144 times 27 and 26 square plus 8/39 square minus 27/26 times 8/39 times. And now we have two times call sign of 18 -18 T. is negative 10 T-. And then close here to Square Bracket -25 144 over 65 sq sign square of five T. Now remember that co sign of eggs. Go sign of any angle is and even function that is co sign of negative arguments equal to the design of the same argument with positive signs. So this is equal to and now we're gonna at the same time take Common factor out of 144 because we have it here so we can take it out of all the whole suppression. So you get this this term plus this term and now we get minus 27 or 26 Times 8/39 times two. Co sign of TNT because co sign is uneven, function -25 times 144 Over 65 square Signs Square of five T Okay, so We had this remember we have taken a common factor of 144 of the whole expression because we have one of those factors here so we can one left inside and the manufacturer now we're gonna use again these expressions here above right here to find the coastline of the angle of the double of an angle. So we get this first expression here using this plus sign, we'll get that co sign of alpha plus alpha. The east coast sign of two Alpha will be equal to the angles are the same. So we get two times go sign of alpha minus. Sorry, I made a mistake here. Both yet. What kind of Alpha times Sine of alpha. So yet call sign square of alpha minus sine square above. We are interesting as using the signs square evolve of alpha because we have some square here. All right, so this replacing this inspiration, we get one minus sine squared of alpha minus and square of alphas minus two sine square of alfa. So this identity we're going to use. Yeah, that is I'm going to put it again here, Co sign of two times out for equal 1 -2 sine squared of health. And we are going to apply that for alpha equals five T. And with that we can say that is equal to we continue here. Yeah, 144 times these same terms here -27. Number 26 Times 8/39 times two times Now a real place go sign of TNT, which is two times five T. by these formulas 1 -2 Sine Square of five T. That's it because go sign of two times five T. There is concern of 20 here. It's replaced by discretion Which is finally 1 -2 cents were of five T. So we are just taking α five T. And using this equality. So we have done almost two. Okay, he's left his term. Finally sign square five t. Remember the square bracket is around here somewhere. Closing here. The square bracket. Okay, So we have 144 times And now we distribute here inside the parentheses. So we get 27, number 26 square Plus 8/39 square -27/26 times 8/39 Times II, which is the first distribution here, Times one. Now we have glass, 27/26 times eight over 39 times two times two because he had this factor to here and this one here signs square of five G. In the last er -25 Over 65 square 10: 144. Sign square of 15. Close yeah Square brackets. So it's equal to 144 times. And now there is interesting thing here, these three terms Our perfect square because we have a square of the number plus the square of the number and -2 times the product of those numbers. So this is actually equal to 27/26 8/39 square. Okay, so now this term here and this last term here has come factor of science square of five T. So we get plus Parenthesis, then we have 27/26 times 8/39 times four, which is two times two -25 times 144. or 65 square times signed Square of five T. She's become a factor and this is it. Now we're going to calculate all these inside these parenthesis And we had 144 times 27 or 36 uh minus 8/39 square plus. I will have here. The denominator could be 26 times 39 Times 65 sq without taking. Uh huh. Key. Uh The common denominator, we are using all the eliminators because we're going to see something beautiful here. And if we do this then in the numerator where we get 20 s 27 times eight times four times 65 square minus. Yeah. Finance 26 39 25. 144 Times Sine Square of five T. Okay, so we're going to do both calculations these calculation inside parenthesis in the first term inside square bracket is we have a common denominator again. We do it without taking too much care. There are Hover away. So it's 26 times 39 times 27 Times 39 eight times 26 and all that square. It's his first term plus. And now here I'm not gonna to the denominator is Britain same. But in the numerator we shouldn't calculator. We calculated to terms 27 times satan four times 66 65, 3 square all that give us his number. Um This number three millions 6 100 50,000 400. It's a big number ever. Not problem with that because When we do these all the calculations, 2016, 39, 10 25 times 144, we get the same number in that. Sorry, I made a mistake. It's important and you have the correct number. So here is not this number. It's sure there is. No there So it's 3650 400. So he's here, that's it and here we get the same and that's the key thing here because the term where we have the chief Parameter T disappears because this is equal to zero. And that's a key thing here to end up with the result of 100 And then here we have in the numerator we get 1053 minus that is 27 times 39. Then eight times 26 is 208 over 26 times 39 and all that square. Yeah, now Is equal to 144 times Now, we simplify the numerator, we get 845 in denominator. The product 26 and 30 90s, 1014 of the square. We recognize that 144 is 12 square. So we had 12 times 845 over 1014 sq. All that square. Mhm. And now we factor out all the terms appear that appear in the Inspiration 12 times 845 over 1014. So again, uh 12 is 4, 10, 3 and four is two times two times three. So it's 12 and 145 Is factored out as five times 13 times 13 and the eliminator for 1,014, which we could have a factor here, but there's no problem. Do it again. Here is two times three times 13 time 13. That is because three times 13 is 39, which is this factor here and two times 13 is 26 jesus factor here. Okay. You know, we cancel out these two, these two History with history and the 2 13th in both numerator and denominator. And we will get this square here, which is the square here. And so this is in the numerator, we get 10 in the denominator. We get one because all is simplified to stand square. He's one hug. So we ended up proving that any point in the curve given parametric lee by these three equations satisfy the question given here in X, Y Z, which is the hyper employed of one shit