How to show express $y $ in terms of angle $\theta$?

Question

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$ABC$ is a straight line with $AB = BC = 3$ units. $B$ is the centre of the circle with radius of $2$ units. $P$ is a point on the circle. $\widehat{B_1} = \theta$, $\widehat{A} = x$, $QC \perp AC$ and $QC = y$.

Show that $$y = \frac{12\sin\theta}{3 + 2\cos\theta}$$.

How do I go about doing this? As you can see, I worked out as much as I can from the diagram and $$\sin\widehat{P_1} = \frac{3}{2} \times \sin x$$ I used the sine rule to work that out.

How do I get the required answer? Many thanks

Arpan · Answer 1 · 2015-05-16T09:23:07.157

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Hint: You can use coordinate-geometry to get your answer. Put $B$ as the origin and $ABC$ as $x$-axis. Then $P\equiv(2\cos\theta,2\sin\theta)$

Can you take it from there?

edited May 16 '15 at 09:23

answered May 16 '15 at 09:22

Arpan

2,633

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Is there no other way? Perhaps using euclidean geometry. I am not sure where to go using your answer. – Aspiring Mathlete May 16 '15 at 09:22
This is the method that first came to head and it seems easy too, that is, not much calculation. There probably exists a pure geometry solution as well, but I can't see it yet. I'll let you know if I get anything. – Arpan May 16 '15 at 09:25
How do you do it according to your solution? I worked out $AP^{2}$ and $PQ^{2}$ separately. – Aspiring Mathlete May 16 '15 at 09:37
@AspiringMathlete Equation of $CQ$ is $x=3$, and you can write equation of $APQ$ because you have two points. Solve to find $Q$, then find $y$. – Arpan May 16 '15 at 09:41
so why do we need to know what $P$ is ? – Aspiring Mathlete May 16 '15 at 09:44
So you can write equation of line $APQ$, you know points $A$ and $P$ – Arpan May 16 '15 at 09:45
I am still struggling: I got the line as $ y = \frac{2\sin \theta}{2\cos \theta + 3 } \times x +c$ – Aspiring Mathlete May 16 '15 at 09:47
First find $c$ by putting $x=-3$,i.e. point $A$. then substitute $x=3$ into the equation and get the y-coordinate of $Q$, which is same as the $y$ in the question. – Arpan May 16 '15 at 09:48
Alright thanks it worked. But now I am trying to find a way without using co-ordinate geometry. – Aspiring Mathlete May 16 '15 at 09:56

How to show express $y $ in terms of angle $\theta$?

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