Y is the center of a circle having radius r. Point X is located outside the circle and tangents XP and XC are drawn to touch the circle respectively at P and C.

Given that XY = d, determine the length of PC in terms of r and d.

let Y be at the origin, then the equation for the circle is given by

x^2+y^2=r^2 and the slope of a tangent line on this circle at (p,q) is -p/q

without loss of generality we can place X at (d,0)  so now we want to find (p,q) such that

(1) p^2+q^2=r^2 and (2) q/(p-d)=-p/q

from (2) we get q^2=p(d-p)=dp-p^2

thus

q^2+p^2=r^2=dp

thus

p=r^2/d

thus q=+-r*sqrt(d^2-r^2)/d

thus the to tangent points are at

(r^2/d, +-r*sqrt(d^2-r^2)/d)

and the desired distance is

4*r^2*sqrt(d^2-r^2)/d

Posted by Daniel on 2007-11-11 00:21:19