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FP2 polar co-ords understanding question

June 2013 Q8, here is the paper; http://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2013/Exam%20materials/6668_01_que_20130621.pdf page 24 of the pdf

and my working;

june2013.jpg

what is the significance of the sin⁡θ=0 \displaystyle \sin\theta = 0 and sin⁡θ=−23 \displaystyle \sin\theta = -\sqrt\frac{2}{3} ?

do they tell you anything about the plot or do you just neglect them by realising the angle required is roughly equal to π4 \displaystyle \frac{\pi}{4}

(kind of like how in mechanics projectiles when using s=ut+12at2 \displaystyle s = ut + \frac{1}{2}at^2 to find t you sometimes get the minus t value which you usually ignore)
dont think it has any real significance in this case, other than that the curve also meets the half line at the origin
Original post by DylanJ42

do they tell you anything about the plot or do you just neglect them by realising the angle required is roughly equal to π4 \displaystyle \frac{\pi}{4}


Although it's not obvious from the diagram, sin⁡θ=0\sin \theta = 0 gives a point where the tangent is parallel to the initial line, but it's actually the origin. We discard it as the requirement is P is distinct from O.

The other value of theta will also give rise to a line parallel to the initial line, but it's outside the domain of the definition of C, so we can again discard it.
(edited 8 years ago)
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Original post by drandy76
dont think it has any real significance in this case, other than that the curve also meets the half line at the origin


PRSOM

Original post by ghostwalker
Although it's not obvious from the diagram, sin⁡θ=0\sin \theta = 0 gives a point where the tangent is parallel to the initial line, but it's actually the origin. We discard it as the requirement is P is distinct from O.

The other value of theta will also give rise to a line parallel to the initial line, but it's outside the domain of the definition of C, so we can again discard it.


i've never realised that the tangent was parallel to initial line at the origin ever, i always thought it was a mysterious recurring solution which we ignored "just because" and i never understood why :laugh:, but this all makes complete sense now

thank you both :biggrin:
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If r→0 r \rightarrow 0 as θ→a \theta \rightarrow a then θ=a \theta = a is a tangent to the curve.
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Original post by B_9710
If r→0 r \rightarrow 0 as θ→a \theta \rightarrow a then θ=a \theta = a is a tangent to the curve.


is this what is happening at the origin?

if i was to zoom in really close...

fpr.jpg

would θ=π2 \displaystyle \theta = \frac{\pi}{2} be a solution to dxdθ \displaystyle \frac{dx}{d\theta} with that same reasoning
(edited 8 years ago)
Original post by DylanJ42
is this what is happening at the origin?

if i was to zoom in really close...

fpr.jpg

would θ=π2 \displaystyle \theta = \frac{\pi}{2} be a solution to dxdθ \displaystyle \frac{dx}{d\theta} with that same reasoning


yeah, comes up a lot in OCR as a precursor to finding area bound by the curve, since the tangents at the origin can be taken as the limits
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Original post by drandy76
yeah, comes up a lot in OCR as a precursor to finding area bound by the curve, since the tangents at the origin can be taken as the limits


exactly yea you're right, i always know to take limits from 0 to pi/2, so i guess it follows that tangents exist at 0 and pi/2

Spoiler

Original post by DylanJ42
exactly yea you're right, i always know to take limits from 0 to pi/2, so i guess it follows that tangents exist at 0 and pi/2

Spoiler



Dw about it just probs not covered in your syllabus, for instance I never know what's going on when people mention conics


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Original post by drandy76
Dw about it just probs not covered in your syllabus, for instance I never know what's going on when people mention conics
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yea i guess, i know it now and that's the main thing :biggrin: thanks again :cute:
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Original post by drandy76
Dw about it just probs not covered in your syllabus, for instance I never know what's going on when people mention conics


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:zomg:

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