# A Level Further Maths Argand Diagrams

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#1
How would you find find a complex number (in the form a + bi) with the smallest possible argument that intersects a circle?
0
6 months ago
#2
(Original post by Toast210)
How would you find find a complex number (in the form a + bi) with the smallest possible argument that intersects a circle?
Can you post the full question?

At a guess, it probably wants you to compute the tangent to the circle, which you could do using quadratic discriminants, trig or pythagoras/similar triangles. Really need to see the question though.
Last edited by mqb2766; 6 months ago
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#3
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6 months ago
#4
(Original post by Toast210)
Ok, have you sketched it? The numbers make a Pythagoras-similar triangles the easy way, but use a tangent point - quadratic discriminant if you're not confident with the geometry behind it.
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#5
Ok I can see a 3,4,5 triangle using the center of the circle and the point of intersection from the origin. I got the modulus of the complex number as 4. Then I would have to take away 2 angles in order to get the argument. Thank you.
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6 months ago
#6
(Original post by Toast210)
Ok I can see a 3,4,5 triangle using the center of the circle and the point of intersection from the origin. I got the modulus of the complex number as 4. Then I would have to take away 2 angles in order to get the argument. Thank you.
I presume you finished it off? Did you notice anything about the side ratio of y : x : 4?
If you're interested, there is a neat bit of (relatively simple, but not necessarily intuitive) geometry underpinning the question.
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#7
(Original post by mqb2766)
I presume you finished it off? Did you notice anything about the side ratio of y : x : 4?
If you're interested, there is a neat bit of (relatively simple, but not necessarily intuitive) geometry underpinning the question.
Yes I managed to finish it thank you. What's this neat bit of geometry if you don't mind me asking?
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6 months ago
#8
(Original post by Toast210)
Yes I managed to finish it thank you. What's this neat bit of geometry if you don't mind me asking?
What did you get for the answer?
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#9
3.84 + 1.12i
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6 months ago
#10
(Original post by Toast210)
3.84 + 1.12i
or as a ratio for y : x : 4
7 : 24 : 25
Notice anything?
Spoiler:
Show
Your original triplet a : b : c was 3 : 4 : 5, so this triplet is
b^2 - a^2 : 2ab : c^2

You could "write down" the answer as
24*4/5^2 + i*7*4/5^2
Last edited by mqb2766; 6 months ago
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