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circle_tangent_intersect_proof

# circle_tangent_intersect_proof - Proof of an implicit line...

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Proof of an implicit line equation through a point (not on the circle) that is tangent to a general circle Derivation of an equation that calculates the the general equation of an implicit line through a point (not on the circle) that is tangent to a general circle: P ( x, y ) J ( x j , y j ) r C ( x c , y c ) 1 We wish to find the implicit equation of the tangent ax + by + c = 0 The coefficients a and b are obtained from: a = r ( x c - x j ) - ( y c - y j ) p ( x c - x j ) 2 + ( y c - y j ) 2 - r 2 ( x c - x j ) 2 + ( y c - y j ) 2 b = r ( y c - y j ) + ( x c - x j ) p ( x c - x j ) 2 + ( y c - y j ) 2 - r 2 ( x c - x j ) 2 + ( y c - y j ) 2 (1) c then obtained from the fact that the tangent passes through J : c = - ax j - by j See http://www.netsoc.tcd.ie/ jgilbert/maths site/applets/circles/tangents to circles.html for a java applet demo. Proof: 1

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P ( x, y ) θ J ( x j , y j ) r C ( x c , y c ) 1 We know the coordinates of the circle ( x c , y c ) and its radius, r . We also know the coordinates of the general point, J ( x j , y j ). We can therefore form a vector t between C and J , t = ( x t , y t ) where x t = x c - x j and y t = y c - y j
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circle_tangent_intersect_proof - Proof of an implicit line...

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