Finding a Point on a CircleDate: 5/28/96 at 16:27:40 From: Anonymous Subject: Finding a point on a circle Given a point on the circumference of a circle P(x0,y0), the center of the circle C(a,b), and the X coordinate of another point on the circumference of the circle P(x1,y1), how do I find the y1 value? I already know x0, y0, a, b, and x1, and the Radius R, just not y1. This is as far as I've gotten: I've used the formulas x = R cos(t) + a to figure out the value of t: (x-a)/R = cos(t) Then I tried to use y = R sin(t) + b to calculate Y, but the value doesn't appear to be correct. Thanks in advance, Joel Date: 5/28/96 at 19:45:3 From: Doctor Pete Subject: Re: Finding a point on a circle A convenient way to approach this problem is by using Cartesian coordinates. Since P(x0,y0) is on C, the distance between P(x0,y0) and the center (a,b) is the radius. Then the equation for C is: (x-a)^2+(y-b)^2 = (x0-a)^2+(y0-b)^2. Substituting (x1,y1), we get: (y1-b)^2 = (x0-a)^2+(y0-b)^2-(x1-a)^2, and solving for y1, we have y1 = b +/- sqrt((x0-x1)(x0+x1-2*a)+(y0-b)^2). Intuitively, this solution makes sense because given some x1, we are looking for the intersection of the line x = x1 with the circle C, which in general will have two solutions. In the cases where the line is tangent to C and thus has only one solution, it is clear that y1 = b, corresponding to the plus/minus term in the above expression being 0. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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