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Unformatted text preview: Proof of Intersection of a Line and a Circle P 1 ( x 1 , y 1 ) P 2 ( x 2 , y 2 ) r C ( x c , y c ) This problem is most easily solved if the circle is in implicit form: ( x- x c ) 2 + ( y- y c ) 2- r 2 = 0 and the line is parametric: x = x + ft y = y + gt Substituting for (parametric line) x and y into the circle equation gives a quadratic equation in t : Two roots of which give points on the line where cuts the circle. t = f ( x c- x ) + g ( y c- y ) p r 2 ( f 2 + g 2 )- ( f ( y c- y )- g ( x c- x )) 2 ( f 2 + g 2 ) The roots maybe coincident if the line is tangential to the circle. P 1 ( x 1 , y 1 ) r C ( x c , y c ) If roots are imaginary then there is no intersection. 1 Proof: The circle is in implicit form: ( x- x c ) 2 + ( y- y c ) 2- r 2 = 0 (1) and the line is parametric: x = x + ft y = y + gt (2) Substituting for (parametric line) x and y from Eqn. 2 into the circle equation, Eqn. 1, gives a quadratic equation in t gives: ( x + ft- x c ) 2 + ( y + gt- y c ) 2- r...
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- Winter '11