WEBVTT
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in this problem. Want to graft the curve with
2
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parametric equations. X equal 27/26 times sign of 80
3
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-8/39 sign of 18 T. Y equals negative 27/26
4
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co sign affects eight T Plus 8/39 co sign of
5
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agency. And Z equal 144 over 65. sign
6
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of five T. Or a parameter T. In
7
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the real numbers. Any party. We show that
8
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the curve lies on the hyper Beloit of one sheet
9
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144 X square plus 144. White square-25.
10
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Z sq Equal 100. So in part a we
11
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are we have sketched a graph of the function is
12
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in a computer program in this case metal up we
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have found is Lying this curve in R. three
14
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for a parameter T. Between negative pi over 8.3
15
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and pi over eight 0.3. Mhm. So in
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this case we see that the craft start at this
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point and it uh goes up this way. And
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if you look at the graph from above we will
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see it on the plane plane. Xy we have
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a sort of circle but not an exact circle.
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Yeah because of the equations of X. Y.
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And then the value of C. Is Increasing when
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sign of five T. is increasing after that.
24
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It is possible that the curve decrease again because we
25
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will have decreasing value of sine of T. And
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so on. So This sketch of the graph that
27
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in three D. He's not not that is to
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see But this case can imagine this care of is
29
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starting at a certain level. And then he's getting
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up up this hill making a sort of around in
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the xy plane and then moving up on the seaplane
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. Yeah. So this is the sketch of the
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graph and now barbie, we're going to calculate all
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this expression here on the left And we will simplify
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all that and we will we will get at the
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end of 100 with that. We have proved that
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the careful eyes on the hyperbole Lloyd of one sheet
38
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. So let's see that started by 144 times because
39
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we have 1 to 44 in both this term and
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this one we're going to put it as a common
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factor for tambien. Then we have that factor times
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X squared plus y square. So we're gonna square
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these two expression here. So we get this case
44
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. Mhm. But scores the first term 27/26 square
45
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times sine square of 80. This is the square
46
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of this term here. Mhm. And-2 times
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the part of the two terms. So minus two
48
00:04:05.800 --> 00:04:15.990 A:middle L:90%
times 27 Over 26 times eight Over 39 times the
49
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product of the sign of 80 times the sine of
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agent E. And now plus the square of the
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second term. 8/39 square times sine square of agency
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. So this is the first term. But I'm
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going to do the other one. So this is
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not close yet. That all these expression from here
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to here represents eggs square. Now we're gonna do
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y square. So we have plus negative 27/26 square
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go science squared of eight T. Then we have
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minus because we have these terms here with different signs
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. So it will end up being negative two times
60
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the product of the two terms At least 27/26 Times
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eight or 39 times goes on of eight T.
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Times go sign of agency. Now plus the square
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of the second term. That is 8/39 square Times
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Co Sign Square of 18 T. And these represents
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all these is why square and now all that plus
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or minus. Because we have in mind is here
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to see that's it. That is this minus year
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. So we get minus continue here 25 Times C
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sq is 140 for over 65 square times Sine square
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of five T. Okay. Means that we are
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here calculating all these expressions is the left hand side
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of this equation over here. So let's put now
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uh, this is Z square. So what we
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are calculating here is 144 x square plus. 144
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way square minus 25. Yes, 25 c square
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. And we will continue here equal 144 times.
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And now we can mm hmm. Combined this term
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with this one Because we have a common factor 27/26
79
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square And it will be a common factor of science
80
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square of 80 plus. Co signers square of 80
81
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. And we know that that is equal to one
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because we have the same angle 80 and the sum
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of the square sign and co signed for the same
84
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angle is equal to one. So We will get
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here 27/26 square. That is grouping this or combining
86
00:08:35.659 --> 00:08:39.080 A:middle L:90%
his term with the swan. No, we did
87
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the same for the last two terms and for the
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00:08:45.279 --> 00:08:50.639 A:middle L:90%
middle terms here and here. Wilder The Middle Times
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00:08:50.639 --> 00:08:54.429 A:middle L:90%
1st and I can see that we have a common
90
00:08:54.429 --> 00:09:01.389 A:middle L:90%
factor 27/26 times 8/39. I'm gonna less the two
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inside after. We take a common factor out and
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the negative time will be taking out too. So
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00:09:09.490 --> 00:09:18.389 A:middle L:90%
we get negative 27/26 times eight Over 39 times.
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Okay, so we get to He's factor too well
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, let's hear inside sign a of 80 times sine
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00:09:31.320 --> 00:09:43.610 A:middle L:90%
of 18 T plus because we have taken the common
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factor, negative factor out Plus Coz I two times
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factor to here with this inside two time. Uh
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goes on Pg Times cold sign of eight and T
100
00:10:13.779 --> 00:10:16.769 A:middle L:90%
. And we close zebra. The curly braces here
101
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because we had this some as multiply by this.
102
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Fresh. Yeah. Now we have blasts and we
103
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do the same for this two terms, this one
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here and this one here and again we have the
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common factor 8/39 square in that multiplied by some of
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Science square of 18 T. And go science square
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of 18 T. And we use the automatic identity
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00:10:48.009 --> 00:10:52.610 A:middle L:90%
here again and we have one inside the branches.
109
00:10:52.610 --> 00:10:56.399 A:middle L:90%
So we get finally, Is the term simplified to
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8/39 square. Gosh, this Her bracket here and
111
00:11:05.009 --> 00:11:07.730 A:middle L:90%
then we have the last term which would be this
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00:11:07.730 --> 00:11:16.149 A:middle L:90%
1-1 25 times 144 or 65 sq sign square
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of five T. Now here we are going to
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00:11:24.379 --> 00:11:30.559 A:middle L:90%
use uh well non formulas for from two economic tree
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00:11:30.570 --> 00:11:33.399 A:middle L:90%
because we're going to simplify these two terms, We
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know that the co sign of the sum of two
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00:11:37.019 --> 00:11:48.350 A:middle L:90%
angles is equal to cool, sign of the first
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angle times go sign of the second, one minus
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sign of the first angle, times Sign of the
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00:12:01.669 --> 00:12:09.710 A:middle L:90%
2nd 1. For the difference of the angles we
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have a change of sign just right here. So
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with this we can conclude that okay, if we
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adopt is to the left hand side and right hand
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side of these equations we get co sign of alpha
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00:12:31.779 --> 00:12:41.549 A:middle L:90%
plus beta Blasco sign of alpha minus vera. Okay
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00:12:41.740 --> 00:12:46.519 A:middle L:90%
, what we get here we get are adding the
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00:12:46.519 --> 00:12:50.639 A:middle L:90%
two expressions, the terms with different signs will cancel
128
00:12:50.639 --> 00:12:56.480 A:middle L:90%
out. So we get two times go sign of
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00:12:56.490 --> 00:13:05.100 A:middle L:90%
alpha time scope, sign of bitter and if we
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do it's this direction, co sign of alpha,
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00:13:11.539 --> 00:13:18.659 A:middle L:90%
minor speed minus co sign of alpha blockers, Beta
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we'll get mm to sign of alpha sign of bid
133
00:13:31.639 --> 00:13:37.070 A:middle L:90%
. So we have used these two expressions here to
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00:13:37.070 --> 00:13:41.340 A:middle L:90%
find these quality is here and that's what we want
135
00:13:41.340 --> 00:13:46.029 A:middle L:90%
because we had on the right side of both equations
136
00:13:46.029 --> 00:13:48.710 A:middle L:90%
we have the products of sine of angles and the
137
00:13:48.710 --> 00:13:52.889 A:middle L:90%
parents of sine of angles which we have in this
138
00:13:52.889 --> 00:14:03.309 A:middle L:90%
expression So if we continue here is equal to we
139
00:14:03.309 --> 00:14:18.029 A:middle L:90%
continue here is equal to 144 times 27 or 26
140
00:14:18.039 --> 00:14:22.169 A:middle L:90%
square we're going to put beside this term, we're
141
00:14:22.169 --> 00:14:28.659 A:middle L:90%
going to put this constant term here plus 8/39 square
142
00:14:30.539 --> 00:14:35.940 A:middle L:90%
and now we have negative or minus 27, times
143
00:14:35.950 --> 00:14:45.460 A:middle L:90%
eight over 39 and inside the curly braces we have
144
00:14:45.840 --> 00:14:48.620 A:middle L:90%
to sign of 80 times and 18 to is his
145
00:14:48.620 --> 00:14:56.039 A:middle L:90%
formula here, second one And we have the factor
146
00:14:56.039 --> 00:14:58.269 A:middle L:90%
to that's what we left factor two in each case
147
00:15:00.240 --> 00:15:03.820 A:middle L:90%
To accurately the left side of the equation and alpha
148
00:15:03.820 --> 00:15:07.659 A:middle L:90%
is equal in this case to 80 Mbira is agency
149
00:15:09.340 --> 00:15:11.179 A:middle L:90%
. So we'll get that this first fresh expression is
150
00:15:11.179 --> 00:15:28.179 A:middle L:90%
go sign of 80 eight T-18 T plus my
151
00:15:28.179 --> 00:15:33.850 A:middle L:90%
necessary because we have a minus here co sign of
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00:15:33.860 --> 00:15:39.799 A:middle L:90%
something that is eight T plus 18 t and now
153
00:15:39.799 --> 00:15:45.179 A:middle L:90%
we convert the other expression that is to Co sign
154
00:15:45.179 --> 00:15:48.710 A:middle L:90%
of 80 plus times. Come sign of 18 t
155
00:15:48.710 --> 00:15:52.669 A:middle L:90%
. You since his first equation here. And so
156
00:15:52.669 --> 00:16:03.350 A:middle L:90%
we get plus co sign of 80 plus 18 T
157
00:16:06.440 --> 00:16:17.149 A:middle L:90%
. Plus plus co sign of a T minus agency
158
00:16:18.940 --> 00:16:22.059 A:middle L:90%
. And we close the converse is because we are
159
00:16:22.059 --> 00:16:27.179 A:middle L:90%
done here and before that we have we had the
160
00:16:27.190 --> 00:16:32.509 A:middle L:90%
custom sir we put it here so we closing square
161
00:16:32.509 --> 00:16:42.610 A:middle L:90%
bracket And-25 144 over 65 sq science square of
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00:16:42.620 --> 00:16:51.590 A:middle L:90%
hefty. Now we can see that some terms cancel
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00:16:51.590 --> 00:16:56.679 A:middle L:90%
out here And inside the killer braces we have uh
164
00:16:56.690 --> 00:17:00.460 A:middle L:90%
these two terms our sex the same with opposite signs
165
00:17:00.460 --> 00:17:03.829 A:middle L:90%
. So they cancel out and these two terms this
166
00:17:03.829 --> 00:17:10.529 A:middle L:90%
one and this one are the same. And so
167
00:17:10.539 --> 00:17:18.549 A:middle L:90%
this is equal to 144 times 27 and 26 square
168
00:17:18.549 --> 00:17:33.170 A:middle L:90%
plus 8/39 square minus 27/26 times 8/39 times. And
169
00:17:33.170 --> 00:17:38.960 A:middle L:90%
now we have two times call sign of 18-18
170
00:17:38.960 --> 00:17:47.890 A:middle L:90%
T. is negative 10 T-. And then
171
00:17:47.900 --> 00:17:55.640 A:middle L:90%
close here to Square Bracket-25 144 over 65 sq
172
00:17:55.640 --> 00:18:03.420 A:middle L:90%
sign square of five T. Now remember that co
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00:18:03.420 --> 00:18:07.890 A:middle L:90%
sign of eggs. Go sign of any angle is
174
00:18:07.900 --> 00:18:14.339 A:middle L:90%
and even function that is co sign of negative arguments
175
00:18:14.339 --> 00:18:17.380 A:middle L:90%
equal to the design of the same argument with positive
176
00:18:17.380 --> 00:18:21.250 A:middle L:90%
signs. So this is equal to and now we're
177
00:18:21.250 --> 00:18:25.450 A:middle L:90%
gonna at the same time take Common factor out of
178
00:18:25.460 --> 00:18:29.759 A:middle L:90%
144 because we have it here so we can take
179
00:18:29.759 --> 00:18:32.029 A:middle L:90%
it out of all the whole suppression. So you
180
00:18:32.029 --> 00:18:41.329 A:middle L:90%
get this this term plus this term and now we
181
00:18:41.329 --> 00:18:56.150 A:middle L:90%
get minus 27 or 26 Times 8/39 times two.
182
00:18:57.440 --> 00:19:00.970 A:middle L:90%
Co sign of TNT because co sign is uneven,
183
00:19:00.970 --> 00:19:14.440 A:middle L:90%
function-25 times 144 Over 65 square Signs Square of
184
00:19:14.440 --> 00:19:22.240 A:middle L:90%
five T Okay, so We had this remember we
185
00:19:22.240 --> 00:19:26.210 A:middle L:90%
have taken a common factor of 144 of the whole
186
00:19:26.210 --> 00:19:33.420 A:middle L:90%
expression because we have one of those factors here so
187
00:19:33.420 --> 00:19:40.109 A:middle L:90%
we can one left inside and the manufacturer now we're
188
00:19:40.109 --> 00:19:47.450 A:middle L:90%
gonna use again these expressions here above right here to
189
00:19:47.450 --> 00:19:51.160 A:middle L:90%
find the coastline of the angle of the double of
190
00:19:51.160 --> 00:19:55.349 A:middle L:90%
an angle. So we get this first expression here
191
00:19:56.539 --> 00:20:00.599 A:middle L:90%
using this plus sign, we'll get that co sign
192
00:20:00.599 --> 00:20:03.539 A:middle L:90%
of alpha plus alpha. The east coast sign of
193
00:20:03.549 --> 00:20:07.309 A:middle L:90%
two Alpha will be equal to the angles are the
194
00:20:07.309 --> 00:20:11.220 A:middle L:90%
same. So we get two times go sign of
195
00:20:11.220 --> 00:20:21.349 A:middle L:90%
alpha minus. Sorry, I made a mistake here
196
00:20:22.140 --> 00:20:26.130 A:middle L:90%
. Both yet. What kind of Alpha times Sine
197
00:20:26.130 --> 00:20:29.460 A:middle L:90%
of alpha. So yet call sign square of alpha
198
00:20:30.240 --> 00:20:38.549 A:middle L:90%
minus sine square above. We are interesting as using
199
00:20:40.230 --> 00:20:42.930 A:middle L:90%
the signs square evolve of alpha because we have some
200
00:20:42.930 --> 00:20:48.240 A:middle L:90%
square here. All right, so this replacing this
201
00:20:48.240 --> 00:20:51.450 A:middle L:90%
inspiration, we get one minus sine squared of alpha
202
00:20:51.460 --> 00:20:55.440 A:middle L:90%
minus and square of alphas minus two sine square of
203
00:20:55.450 --> 00:20:57.670 A:middle L:90%
alfa. So this identity we're going to use.
204
00:20:59.039 --> 00:21:02.740 A:middle L:90%
Yeah, that is I'm going to put it again
205
00:21:02.740 --> 00:21:06.460 A:middle L:90%
here, Co sign of two times out for equal
206
00:21:06.460 --> 00:21:10.799 A:middle L:90%
1-2 sine squared of health. And we are
207
00:21:10.799 --> 00:21:15.049 A:middle L:90%
going to apply that for alpha equals five T.
208
00:21:18.509 --> 00:21:22.059 A:middle L:90%
And with that we can say that is equal to
209
00:21:22.440 --> 00:21:32.339 A:middle L:90%
we continue here. Yeah, 144 times these same
210
00:21:32.339 --> 00:21:48.269 A:middle L:90%
terms here-27. Number 26 Times 8/39 times two
211
00:21:48.740 --> 00:21:53.759 A:middle L:90%
times Now a real place go sign of TNT,
212
00:21:53.759 --> 00:21:57.160 A:middle L:90%
which is two times five T. by these formulas
213
00:21:57.839 --> 00:22:07.230 A:middle L:90%
1-2 Sine Square of five T. That's it
214
00:22:07.230 --> 00:22:10.640 A:middle L:90%
because go sign of two times five T. There
215
00:22:10.640 --> 00:22:15.769 A:middle L:90%
is concern of 20 here. It's replaced by discretion
216
00:22:15.539 --> 00:22:19.349 A:middle L:90%
Which is finally 1-2 cents were of five T
217
00:22:21.339 --> 00:22:23.579 A:middle L:90%
. So we are just takingα five T.
218
00:22:23.589 --> 00:22:30.220 A:middle L:90%
And using this equality. So we have done almost
219
00:22:30.220 --> 00:22:37.269 A:middle L:90%
two. Okay, he's left his term. Finally
220
00:22:38.440 --> 00:22:45.940 A:middle L:90%
sign square five t. Remember the square bracket is
221
00:22:45.950 --> 00:22:51.450 A:middle L:90%
around here somewhere. Closing here. The square bracket
222
00:22:53.039 --> 00:23:10.190 A:middle L:90%
. Okay, So we have 144 times And now
223
00:23:10.190 --> 00:23:12.309 A:middle L:90%
we distribute here inside the parentheses. So we get
224
00:23:12.319 --> 00:23:23.670 A:middle L:90%
27, number 26 square Plus 8/39 square-27/26 times
225
00:23:23.680 --> 00:23:30.940 A:middle L:90%
8/39 Times II, which is the first distribution here
226
00:23:30.950 --> 00:23:38.150 A:middle L:90%
, Times one. Now we have glass, 27/26
227
00:23:38.150 --> 00:23:48.390 A:middle L:90%
times eight over 39 times two times two because he
228
00:23:48.390 --> 00:23:52.660 A:middle L:90%
had this factor to here and this one here signs
229
00:23:52.660 --> 00:24:00.670 A:middle L:90%
square of five G. In the last er-25
230
00:24:02.839 --> 00:24:11.720 A:middle L:90%
Over 65 square 10: 144. Sign square of
231
00:24:11.720 --> 00:24:18.069 A:middle L:90%
15. Close yeah Square brackets. So it's equal
232
00:24:18.069 --> 00:24:26.529 A:middle L:90%
to 144 times. And now there is interesting thing
233
00:24:26.529 --> 00:24:33.470 A:middle L:90%
here, these three terms Our perfect square because we
234
00:24:33.470 --> 00:24:36.099 A:middle L:90%
have a square of the number plus the square of
235
00:24:36.099 --> 00:24:41.359 A:middle L:90%
the number and-2 times the product of those numbers
236
00:24:41.240 --> 00:24:52.460 A:middle L:90%
. So this is actually equal to 27/26 8/39 square
237
00:24:56.140 --> 00:24:59.289 A:middle L:90%
. Okay, so now this term here and this
238
00:24:59.289 --> 00:25:02.740 A:middle L:90%
last term here has come factor of science square of
239
00:25:02.740 --> 00:25:07.869 A:middle L:90%
five T. So we get plus Parenthesis, then
240
00:25:07.869 --> 00:25:17.569 A:middle L:90%
we have 27/26 times 8/39 times four, which is
241
00:25:17.569 --> 00:25:26.279 A:middle L:90%
two times two-25 times 144. or 65 square
242
00:25:26.289 --> 00:25:32.150 A:middle L:90%
times signed Square of five T. She's become a
243
00:25:32.150 --> 00:25:37.259 A:middle L:90%
factor and this is it. Now we're going to
244
00:25:37.259 --> 00:25:44.190 A:middle L:90%
calculate all these inside these parenthesis And we had 144
245
00:25:44.240 --> 00:26:06.349 A:middle L:90%
times 27 or 36 uh minus 8/39 square plus.
246
00:26:07.440 --> 00:26:12.660 A:middle L:90%
I will have here. The denominator could be 26
247
00:26:12.670 --> 00:26:22.349 A:middle L:90%
times 39 Times 65 sq without taking. Uh huh
248
00:26:23.740 --> 00:26:30.109 A:middle L:90%
. Key. Uh The common denominator, we are
249
00:26:30.119 --> 00:26:33.900 A:middle L:90%
using all the eliminators because we're going to see something
250
00:26:33.910 --> 00:26:37.730 A:middle L:90%
beautiful here. And if we do this then in
251
00:26:37.730 --> 00:26:45.940 A:middle L:90%
the numerator where we get 20 s 27 times eight
252
00:26:45.950 --> 00:26:56.599 A:middle L:90%
times four times 65 square minus. Yeah. Finance
253
00:27:00.940 --> 00:27:21.500 A:middle L:90%
26 39 25. 144 Times Sine Square of five
254
00:27:21.500 --> 00:27:34.039 A:middle L:90%
T. Okay, so we're going to do both
255
00:27:34.049 --> 00:27:40.160 A:middle L:90%
calculations these calculation inside parenthesis in the first term inside
256
00:27:40.160 --> 00:27:45.910 A:middle L:90%
square bracket is we have a common denominator again.
257
00:27:45.910 --> 00:27:52.569 A:middle L:90%
We do it without taking too much care. There
258
00:27:52.579 --> 00:27:59.309 A:middle L:90%
are Hover away. So it's 26 times 39 times
259
00:27:59.319 --> 00:28:08.359 A:middle L:90%
27 Times 39 eight times 26 and all that square
260
00:28:14.240 --> 00:28:18.660 A:middle L:90%
. It's his first term plus. And now here
261
00:28:19.359 --> 00:28:25.809 A:middle L:90%
I'm not gonna to the denominator is Britain same.
262
00:28:29.539 --> 00:28:32.529 A:middle L:90%
But in the numerator we shouldn't calculator. We calculated
263
00:28:32.529 --> 00:28:38.099 A:middle L:90%
to terms 27 times satan four times 66 65,
264
00:28:38.099 --> 00:28:45.269 A:middle L:90%
3 square all that give us his number. Um
265
00:28:48.140 --> 00:28:59.220 A:middle L:90%
This number three millions 6 100 50,000 400. It's
266
00:28:59.220 --> 00:29:00.859 A:middle L:90%
a big number ever. Not problem with that because
267
00:29:02.240 --> 00:29:04.970 A:middle L:90%
When we do these all the calculations, 2016,
268
00:29:04.970 --> 00:29:08.109 A:middle L:90%
39, 10 25 times 144, we get the
269
00:29:08.109 --> 00:29:15.690 A:middle L:90%
same number in that. Sorry, I made a
270
00:29:15.690 --> 00:29:22.000 A:middle L:90%
mistake. It's important and you have the correct number
271
00:29:22.000 --> 00:29:26.569 A:middle L:90%
. So here is not this number. It's sure
272
00:29:26.569 --> 00:29:34.740 A:middle L:90%
there is. No there So it's 3650 400.
273
00:29:34.740 --> 00:29:40.150 A:middle L:90%
So he's here, that's it and here we get
274
00:29:40.150 --> 00:29:45.950 A:middle L:90%
the same and that's the key thing here because the
275
00:29:45.950 --> 00:29:52.000 A:middle L:90%
term where we have the chief Parameter T disappears because
276
00:29:52.009 --> 00:29:56.680 A:middle L:90%
this is equal to zero. And that's a key
277
00:29:56.680 --> 00:30:02.849 A:middle L:90%
thing here to end up with the result of 100
278
00:30:03.039 --> 00:30:07.599 A:middle L:90%
And then here we have in the numerator we get
279
00:30:07.609 --> 00:30:14.950 A:middle L:90%
1053 minus that is 27 times 39. Then eight
280
00:30:14.950 --> 00:30:25.240 A:middle L:90%
times 26 is 208 over 26 times 39 and all
281
00:30:25.240 --> 00:30:33.109 A:middle L:90%
that square. Yeah, now Is equal to 144
282
00:30:33.109 --> 00:30:37.529 A:middle L:90%
times Now, we simplify the numerator, we get
283
00:30:37.539 --> 00:30:44.160 A:middle L:90%
845 in denominator. The product 26 and 30 90s
284
00:30:44.839 --> 00:30:52.240 A:middle L:90%
, 1014 of the square. We recognize that 144
285
00:30:52.240 --> 00:30:57.000 A:middle L:90%
is 12 square. So we had 12 times 845
286
00:30:57.009 --> 00:31:03.170 A:middle L:90%
over 1014 sq. All that square. Mhm.
287
00:31:06.140 --> 00:31:10.940 A:middle L:90%
And now we factor out all the terms appear that
288
00:31:10.940 --> 00:31:18.960 A:middle L:90%
appear in the Inspiration 12 times 845 over 1014.
289
00:31:18.740 --> 00:31:22.589 A:middle L:90%
So again, uh 12 is 4, 10,
290
00:31:22.589 --> 00:31:26.359 A:middle L:90%
3 and four is two times two times three.
291
00:31:27.140 --> 00:31:33.579 A:middle L:90%
So it's 12 and 145 Is factored out as five
292
00:31:33.589 --> 00:31:42.799 A:middle L:90%
times 13 times 13 and the eliminator for 1,014,
293
00:31:42.809 --> 00:31:48.420 A:middle L:90%
which we could have a factor here, but there's
294
00:31:48.420 --> 00:31:51.029 A:middle L:90%
no problem. Do it again. Here is two
295
00:31:51.029 --> 00:32:02.880 A:middle L:90%
times three times 13 time 13. That is because
296
00:32:02.890 --> 00:32:07.420 A:middle L:90%
three times 13 is 39, which is this factor
297
00:32:07.420 --> 00:32:13.359 A:middle L:90%
here and two times 13 is 26 jesus factor here
298
00:32:14.339 --> 00:32:17.690 A:middle L:90%
. Okay. You know, we cancel out these
299
00:32:17.690 --> 00:32:22.390 A:middle L:90%
two, these two History with history and the 2
300
00:32:22.390 --> 00:32:27.490 A:middle L:90%
13th in both numerator and denominator. And we will
301
00:32:27.490 --> 00:32:30.059 A:middle L:90%
get this square here, which is the square here
302
00:32:30.839 --> 00:32:32.690 A:middle L:90%
. And so this is in the numerator, we
303
00:32:32.690 --> 00:32:37.369 A:middle L:90%
get 10 in the denominator. We get one because
304
00:32:37.369 --> 00:32:43.440 A:middle L:90%
all is simplified to stand square. He's one hug
305
00:32:44.240 --> 00:32:50.339 A:middle L:90%
. So we ended up proving that any point in
306
00:32:50.339 --> 00:32:55.529 A:middle L:90%
the curve given parametric lee by these three equations satisfy
307
00:32:55.539 --> 00:33:00.579 A:middle L:90%
the question given here in X, Y Z,
308
00:33:00.589 --> 00:33:02.259 A:middle L:90%
which is the hyper employed of one shit