jungkook123
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#1
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Given that k = p√5 and that p > 0, state the value of p as a fraction in its lowest terms.
The point A lies on the circle with equation x^2+y^2=9 and vector OA = 2ki + kj

so far I substituted k into the vector so its OA=2p√5i+p√5j but then I don't know what to do next.
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*****deadness
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I had a look and I didn't do it by substituting k in right away. What does OA tell you about the relationship between the x and y coordinates of A? Given the circle equation and A lies on it, there's only one point that satisfies both (given k>0). But I suppose the substitution first would get you there as well.
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jungkook123
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(Original post by *****deadness)
I had a look and I didn't do it by substituting k in right away. What does OA tell you about the relationship between the x and y coordinates of A? Given the circle equation and A lies on it, there's only one point that satisfies both (given k>0). But I suppose the substitution first would get you there as well.
in the OA vector, j moves the y coordinate and i moves the x coordinate?
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*****deadness
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Yeah
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jungkook123
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(Original post by *****deadness)
Yeah
if you don't substitute it first, what would you do?
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*****deadness
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Well OA=2ki+kj, so without giving the method away, this directly tells me the ratio between the x and y coordinate of A. Then rewriting y in terms of x at A (or x in terms of y if you like), substitute into the circle equation to find for which point on the circle it has that ratio
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jungkook123
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(Original post by *****deadness)
Well OA=2ki+kj, so without giving the method away, this directly tells me the ratio between the x and y coordinate of A. Then rewriting y in terms of x at A (or x in terms of y if you like), substitute into the circle equation to find for which point on the circle it has that ratio
well I think I have done this completely wrong but I've done so far
let i = x and j = y
2kx+ky=0
ky=-2kx
y=-2kx/k
substitute y into the circle equation
x^2-2kx/k=9
I'm I going in the wrong path?
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*****deadness
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Slightly...since we're going from the origin O, we've moved 2ki+kj to A, which means moving 2k in the positive x direction and k in the positive y direction. And since we started from O, A is (2k,k). Now this can be anywhere since k is a constant, but there is only 1 of these points that also lies on the circle
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jungkook123
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(Original post by *****deadness)
Slightly...since we're going from the origin O, we've moved 2ki+kj to A, which means moving 2k in the positive x direction and k in the positive y direction. And since we started from O, A is (2k,k). Now this can be anywhere since k is a constant, but there is only 1 of these points that also lies on the circle
since k=p√5, what do I do to find what p is? I understand what the question is telling me to do but I'm unsure on how to solve it?
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jungkook123
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(Original post by *****deadness)
Slightly...since we're going from the origin O, we've moved 2ki+kj to A, which means moving 2k in the positive x direction and k in the positive y direction. And since we started from O, A is (2k,k). Now this can be anywhere since k is a constant, but there is only 1 of these points that also lies on the circle
oh wait, considering A is (2k,k) do I substitute x=2k and y=k into the circle equation?
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*****deadness
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Yeah and you'll get what k is which tells you what p is
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jungkook123
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(Original post by *****deadness)
Yeah and you'll get what k is which tells you what p is
so I did
2k^2+k^2-9=0
k=-1+√73/4 or -1-√73/4
considering p>0, k=-1+√73/4
divide k by √5 to find p and i get p=√365-√5/20
it's quite a wacky number so im not sure if im correct?
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*****deadness
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Yeah it's (2k)^2
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*****deadness
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And even with 3k^2-9=0 what went so wrong that you got that? Even with this one it's k=-root 3 or +root 3 right
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*****deadness
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Yeah it's not what I got, which gives me p as an actual fraction
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jungkook123
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(Original post by *****deadness)
And even with 3k^2-9=0 what went so wrong that you got that? Even with this one it's k=-root 3 or +root 3 right
I accidentally did 2k^2+k-9 instead of 3k^2-9 oops
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jungkook123
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#17
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(Original post by *****deadness)
Yeah it's not what I got, which gives me p as an actual fraction
I ended up with p=√15/5?
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*****deadness
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#18
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NO it's actually (2k)^2
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*****deadness
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#19
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I said that because I was confused how you got the ugly surds even with 3k^2-9=0, but it should really be (2k)^2+(k)^2-9=0
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jungkook123
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#20
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(Original post by *****deadness)
I said that because I was confused how you got the ugly surds even with 3k^2-9=0, but it should really be (2k)^2+(k)^2-9=0
finally got it now😭😭, thank you so much for your help
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