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# A basic circle theorem problem watch

1. I'm struggling with a little task I'm trying to do, see if any of you guys can solve this one in basic terms for me:

Two circles overlap such that at the points of intersection, the lines are at right angles. A straight line joins the two points of intersection. Calculate the distance from a point, X, to the straight line along the radial line through the point, which lies on the circumference of the other circle, and the radius of the first circle. The distance must be in terms of the radius, R of the circle whose centre you are using the measrue the distance, and the distance to the point X along the radial line: x.

Here is a sketch:
Attached Images

2. Apologies for being slow, but I don't really understand the question. Or the diagram either.
3. OK, there are two circles, Circle A and Circle B. They overlap. Where they overlap they cross at right angles. There is a random point on the circumference of Circle B, on the section that is enclosed by Circle A. The radius of Circle A = R. The distance to this random point from the circle A centre is x. A line is drawn connecting the centre of Circle A and the random point and continued to cross the straight line between the two circles. Find the distance between the random point and the intersection of the two straight lines. Express it in terms of R and x.
4. I have no idea, sorry
5. OK, I think I understand the problem now, but surely the distance you're trying to find depends on the position of the random point on B's circumference. So shouldn't your answer be a variable (including a sin(theta) or something in there)?
6. no, the answer is definitely expressible in only x and R
7. (Original post by Squishy)
OK, I think I understand the problem now, but surely the distance you're trying to find depends on the position of the random point on B's circumference. So shouldn't your answer be a variable (including a sin(theta) or something in there)?
x is a variable as it's the distance from the centre of A to X which changes depending on where X is on the circumference of B.

I don't have a clue what the answer is though!!
8. (Original post by Bezza)
x is a variable as it's the distance from the centre of A to X which changes depending on where X is on the circumference of B.

I don't have a clue what the answer is though!!
Yeah, I know...it's just for what it's worth, I get the distance to be R/[sqrt(2).sin(theta)] - x, where theta is some angle that's difficult to describe. It's probably wrong anyway...maybe I can find some way to reduce it to just R and x later.
9. yup you're wrong....the answer is something like (2R^2)/(R^2+x^2), or something similar, i cant quite remember.
10. Here is a sketch solution. Please tell me if you disagree with what I have done or want more details.

(0) Make all the definitions suggested by the attached picture. (I prefer to use |OX| rather than x to denote the distance between O and X.)

(1) Use the fact that X is on the second circle to show that

x* = [sqrt(R^2 + S^2) - sqrt(S^2 - R^2 m^2)] / [1 + m^2].

(2) Let k = sqrt(1 + m^2). Show that k = |OX| / x*.

(3) Deduce from (1) and (2) that

k^2 = 4 |OX|^2 (R^2 + S^2) / (R^2 + |OX|^2)^2.

(4) Show that |OC| = R^2 / sqrt(R^2 + S^2).

(5) Show that |OY| = k|OC| = 2 |OX| R^2 / (R^2 + |OX|^2).

(6) Deduce from (5) that

|XY|
= |OY| - |OX|
= |OX| (R^2 - |OX|^2) / (R^2 + |OX|^2).
Attached Images
11. circ-tri2lo.pdf (73.2 KB, 113 views)
12. i'm sorry but i cant even do stage 1, can you expand. I can only get to:

x = sqrt(R^2 - s^2) - sqrt(s^2 - m^2x^2)
13. (Original post by Willla2)
i'm sorry but i cant even do stage 1, can you expand. I can only get to:

x = sqrt(R^2 - s^2) - sqrt(s^2 - m^2x^2)
|OA| = sqrt(R^2 + S^2) because OBA is a right angle.

The point X, with coordinates (x*, mx*), is a distance S from A. So

(x* - sqrt(R^2 + S^2))^2 + (mx*)^2 = S^2,
x*^2 - 2 x* sqrt(R^2 + S^2) + R^2 + S^2 + m^2 x*^2 = S^2,
x*^2 (1 + m^2) - 2 x* sqrt(R^2 + S^2) + R^2 = 0.

Solving the quadratic and taking the smaller root gives

x*
= {2 sqrt(R^2 + S^2) - sqrt[4(R^2 + S^2) - 4 R^2 (1 + m^2)]} / {2(1 + m^2)}
= {sqrt(R^2 + S^2) - sqrt[(R^2 + S^2) - R^2 (1 + m^2)]} / {1 + m^2}
= {sqrt(R^2 + S^2) - sqrt(S^2 - R^2 m^2)} / {1 + m^2}.

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Updated: August 8, 2004
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