Consider the two circles a(t) = (cos(t), sin(t)), B(t) = (2+2 cos(t), 1+2 sin(t)). where t e [0, 27]. In this problem, we will show that the two circles meet at a right-angle at the point Q, as in the below diagram. -34 25 15 05 15 -1 -0.5 10 0.5 1.5 2.5 3.5 4.5 -0.5 (a) Your friend Tonga is trying to find the exact Cartesian coordinates of Q. But she ends up proving that the two circles never actually intersect. Here is her argument: To find the intersection points, we set a(t) = B(t) and solve for t. This results in two equations: cos(t) = 2+2 cos(t) and sin(t) = 1+2 sin(t). But the first equation is equivalent to cos(t) = -2, which has no solution! This means that there is no value of t for which a(t) = B(t), which means that the two circles never intersect. What's wrong with Tonga's argument? 2,
Consider the two circles a(t) = (cos(t), sin(t)), B(t) = (2+2 cos(t), 1+2 sin(t)). where t e [0, 27]. In this problem, we will show that the two circles meet at a right-angle at the point Q, as in the below diagram. -34 25 15 05 15 -1 -0.5 10 0.5 1.5 2.5 3.5 4.5 -0.5 (a) Your friend Tonga is trying to find the exact Cartesian coordinates of Q. But she ends up proving that the two circles never actually intersect. Here is her argument: To find the intersection points, we set a(t) = B(t) and solve for t. This results in two equations: cos(t) = 2+2 cos(t) and sin(t) = 1+2 sin(t). But the first equation is equivalent to cos(t) = -2, which has no solution! This means that there is no value of t for which a(t) = B(t), which means that the two circles never intersect. What's wrong with Tonga's argument? 2,
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Consider the two circles
a(t) = (cos(t), sin(t)),
B(t) = (2+2 cos(t), 1+2 sin(t)).
where t e [0, 27]. In this problem, we will show that the two circles meet at a right-angle at the point
Q, as in the below diagram.
3-
25
15
05
15
-0.5
10
0.5
1.5
2.5
3.5
4.5
-05
(a) Your friend Tonga is trying to find the exact Cartesian coordinates of Q. But she ends up
proving that the two circles never actually intersect. Here is her argument:
To find the intersection points, we set a(t) = B(t) and solve for t. This results in two
equations: cos(t) = 2+2 cos(t) and sin(t) = 1+2 sin(t). But the first equation is equivalent
to cos(t) = -2, which has no solution! This means that there is no value of t for which
a(t) = B(t), which means that the two circles never intersect.
What's wrong with Tonga's argument?
(b) Find the exact Cartesian coordinates of Q by solving the two simultaneous equations r? +y? = 1
and (z – 2)? + (y – 1)² = 4.
(c) At what time does a reach Q? At what time does 3 reach Q?
(Hint: arctan is your friend and your enemy. It only outputs angles between –7/2 and T/2.
But the two angles you're looking for are not in that range .)
(d) Compute the tangent vectors of a and B at point Q and show that they are orthogonal.
(Fun Fact: Since the two circles meet at a right-angle, we say that they intersect transversally.)
2,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faba73b71-75cf-41e8-b53e-fd46ec14d4a7%2F17ab7f64-38dc-461a-b40b-13c1f2ffc75d%2Fy8oa0lo_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the two circles
a(t) = (cos(t), sin(t)),
B(t) = (2+2 cos(t), 1+2 sin(t)).
where t e [0, 27]. In this problem, we will show that the two circles meet at a right-angle at the point
Q, as in the below diagram.
3-
25
15
05
15
-0.5
10
0.5
1.5
2.5
3.5
4.5
-05
(a) Your friend Tonga is trying to find the exact Cartesian coordinates of Q. But she ends up
proving that the two circles never actually intersect. Here is her argument:
To find the intersection points, we set a(t) = B(t) and solve for t. This results in two
equations: cos(t) = 2+2 cos(t) and sin(t) = 1+2 sin(t). But the first equation is equivalent
to cos(t) = -2, which has no solution! This means that there is no value of t for which
a(t) = B(t), which means that the two circles never intersect.
What's wrong with Tonga's argument?
(b) Find the exact Cartesian coordinates of Q by solving the two simultaneous equations r? +y? = 1
and (z – 2)? + (y – 1)² = 4.
(c) At what time does a reach Q? At what time does 3 reach Q?
(Hint: arctan is your friend and your enemy. It only outputs angles between –7/2 and T/2.
But the two angles you're looking for are not in that range .)
(d) Compute the tangent vectors of a and B at point Q and show that they are orthogonal.
(Fun Fact: Since the two circles meet at a right-angle, we say that they intersect transversally.)
2,
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