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Getting To PC With A Circle (Posted on 2007-11-10) Difficulty: 2 of 5
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.

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (2 votes)

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Solution solution (different approach) | Comment 2 of 9 |

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
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