Did you get that the coordinates of M are ? If so, then the next part, you can do by verification.(Original post by RDKGames)
How would I go about b. ii? I found the constant distance to be 2a in prev part but I'm not sure where to begin for this one.
Attachment 566676
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Edit: this is the Cartesian coordinates.
Alternatively, if you do not seek the method of verification, you can do so by doing this:
As we can use the fact that so we get that so we can obtain that
Using the similar approach, notice that Hence, .Using .
Rearranging, Notice for any varying and we can say that .
Finally, notice that . Therefore .
We can cancel the r, aslong as covers the point at the pole i.e at .
This will give us the required result.
Year 13 Maths Help Thread

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 02082016 13:39
Last edited by AMarques; 02082016 at 14:26. 
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 02082016 15:43

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 02082016 16:11
(Original post by AMarques)
Did you get that the coordinates of M are ? If so, then the next part, you can do by verification.
Edit: this is the Cartesian coordinates.
Alternatively, if you do not seek the method of verification, you can do so by doing this:
As we can use the fact that so we get that so we can obtain that
Using the similar approach, notice that Hence, .Using .
Rearranging, Notice for any varying and we can say that .
Finally, notice that . Therefore .
We can cancel the r, aslong as covers the point at the pole i.e at .
This will give us the required result.Last edited by RDKGames; 02082016 at 16:13. 
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 02082016 16:24
(Original post by RDKGames)
Your working out makes sense and it gets the right answer, however I'm unsure how the get the coordinates of M. I'm not used to working with lines in polar form. I can see that the line's equation is but I'm unsure what substitutions go where to get it.
For the polar equation this is the same as in Cartesian form. For we get the + sign for a > 0, and  for a < 0.
We then sub for in the second equation. By doing this, we can find both A and B, and then M will just be the average of the x/y coordinates.
Out of curiosity how did you find the distance AB without finding A and B explicitly?Last edited by AMarques; 02082016 at 16:27. 
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 02082016 16:56
(Original post by AMarques)
I basically turned both equations into Cartesian form, and then subbed one into the other and made x the subject. By doing this, I get three xcoordinates (including the pole which we don't count as either A or B), and hence three ycoordinates.
For the polar equation this is the same as in Cartesian form. For we get the + sign for a > 0, and  for a < 0.
We then sub for in the second equation. By doing this, we can find both A and B, and then M will just be the average of the x/y coordinates.
Out of curiosity how did you find the distance AB without finding A and B explicitly?
For point B (and I'm not entirely sure how to explain this properly), I simply imagined that line at angle above the initial line OL and the angle of clockwise, below OL, would give the second point, therefore B would be .
Since r represents the length, I simply added and : thus giving which is a constant in dependent of that I required.Last edited by RDKGames; 02082016 at 17:34. 
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 02082016 17:37
(Original post by RDKGames)
Your working out makes sense and it gets the right answer, however I'm unsure how the get the coordinates of M. I'm not used to working with lines in polar form. I can see that the line's equation is but I'm unsure what substitutions go where to get it. I can only see M being
Then the coordinates of the midpoint at point M are .
You can convert from parametric to Cartesian equation in and and then back to polar form.Last edited by B_9710; 02082016 at 18:03. 
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 02082016 17:40
(Original post by RDKGames)
To find the distance, I firstly knew that and used that substitution into and that would give me intersection point A as a function of .
For point B (and I'm not entirely sure how to explain this properly), I simply imagined that line at angle above the initial line OL and the angle of clockwise, below OL, would give the second point, therefore B would be .
Since r represents the length, I simply added and : thus giving which is a constant in dependent of that I required.
Edit: The post above explains how to do itLast edited by AMarques; 02082016 at 18:57. 
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 02082016 19:08
(Original post by AMarques)
Oh that is quite a nice way of doing it. However, not sure how using polar coordinates would work to find M. Does what I said previously make sense?
Edit: The post above explains how to do it
I've only started polar stuff yesterday so I do prefer turning everything into Cartesian as of now so I'll keep this method handy, along with my geometric observation as shown. Is there just a method to notice the intersections for A and B straight away or would I necessarily have to do some substitutions before deriving them? 
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 02082016 20:55
(Original post by RDKGames)
Yes it does make sense, thank you.
I've only started polar stuff yesterday so I do prefer turning everything into Cartesian as of now so I'll keep this method handy, along with my geometric observation as shown. Is there just a method to notice the intersections for A and B straight away or would I necessarily have to do some substitutions before deriving them?
Glad I was helpful, let me know how the polar coordinates stuff goes. 
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 02082016 21:03
(Original post by AMarques;[url="tel:66724118")
66724118[/url]]Using polar equations substitutions is normally the easiest method as demonstrated above, however there is no harm in doing Cartesian. When you say "notice the intersections" what do you mean? I normally sketch the given curves and it becomes apparent where they occur. In this case you could just substitute and , but in more common polar coordinates questions you'll get, you will normally get equations of the form (of course they can make it more complicated). In this case, you just substitute for and find the corresponding and so on...
Glad I was helpful, let me know how the polar coordinates stuff goes.Last edited by RDKGames; 02082016 at 21:04. 
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 02082016 21:11
(Original post by RDKGames)
Ah yeah, thanks. I simply meant whether there is a fast way to get the points of interaction just by inspection. I'll post more polar stuff here which I may not understand so I can give you a tag if you're able to help further on this topic 
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 03082016 11:34
Towards the end when it says "assuming that I has been found, the differential equation becomes..."
Where do they pluck that from? If I is as they define it there as e to the integral of P(x), where do they make that substitution? I am failing to understand how they get that line
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 03082016 12:15
(Original post by RDKGames)
Towards the end when it says "assuming that I has been found, the differential equation becomes..."
Where do they pluck that from? If I is as they define it there as e to the integral of P(x), where do they make that substitution? I am failing to understand how they get that line
Posted from TSR MobileLast edited by AMarques; 03082016 at 12:16. 
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 03082016 12:20
(Original post by RDKGames)
Towards the end when it says "assuming that I has been found, the differential equation becomes..."
Where do they pluck that from? If I is as they define it there as e to the integral of P(x), where do they make that substitution? I am failing to understand how they get that line
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Indeed, we want to choose an such that is the LHS of our differential equation. Differentiate that using the product rule to get .
Once we have that, we're done since we get and then divide both sides by .
Notice how this almost looks like our original differential equation multiplied through by ? Infact, the only thing that's different is that the coefficient of our term is here and there. Since we want to choose such that and hence allowing us to solve our differential equation, we need to make those two coefficients equal.
i.e: .
That is the motivating thing to do here is to take a DE, multiply it through by a certain function that turns it into an exact differential equation; to find this function, it must satisfy the above. 
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 03082016 13:05
(Original post by AMarques)
...(Original post by Zacken)
....
Ah thank you very much for the explanations. After spending much time and getting confused due to all the P's, Q's and I's, I managed to understand this finally. Had to prove to myself that which made the substitution clear as you've mentioned. 
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 03082016 13:21
(Original post by RDKGames)
Ah thank you very much for the explanations. After spending much time and getting confused due to all the P's, Q's and I's, I managed to understand this finally. Had to prove to myself that which made the substitution clear as you've mentioned. 
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 03082016 15:11
The answer for 2.b) is 196 but i got 256
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 03082016 15:17

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 03082016 15:19
(Original post by B_9710)
256 is correct. 
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 03082016 15:21
Last edited by RDKGames; 03082016 at 15:22.
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