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Finding a Point on a Circle

Date: 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,

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/   
Associated Topics:
High School Conic Sections/Circles
High School Coordinate Plane Geometry
High School Equations, Graphs, Translations
High School Geometry

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