document.write( "Question 1085694: Obtain the condition that lx+my+n =0 may be tangent to the circle x^2+y^2+2gx+2fy+c=0 \n" ); document.write( "

Algebra.Com's Answer #699794 by Fombitz(32388) $\"\"$ $\"About$

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Let the point of intersection be (u,v).
\n" ); document.write( "So then,
\n" ); document.write( " $\"u%5E2%2Bv%5E2%2B2gu%2B2fv%2Bc=0\"$
\n" ); document.write( "and
\n" ); document.write( " $\"lu%2Bmv%2Bn=0\"$
\n" ); document.write( "You know that the slope of the tangent line is equal to the value of the derivative at the intersection point.
\n" ); document.write( "Find the derivative using implicit differentiation,
\n" ); document.write( " $\"2xdx%2B2ydy%2B2gdx%2B2fdy=0\"$
\n" ); document.write( " $\"%282x%2B2g%29dx%2B%282y%2B2f%29dy=0\"$
\n" ); document.write( " $\"%28y%2Bf%29dy=-%28x%2Bg%29dx\"$
\n" ); document.write( " $\"dy%2Fdx=-%28x%2Bg%29%2F%28y%2Bf%29\"$
\n" ); document.write( "So then at (u,v),
\n" ); document.write( " $\"m=dy%2Fdx=-%28u%2Bg%29%2F%28v%2Bf%29\"$
\n" ); document.write( "Using the point slope form of a line,
\n" ); document.write( " $\"y-v=-%28%28u%2Bg%29%2F%28v%2Bf%29%29%28x-u%29\"$
\n" ); document.write( " $\"%28y-v%29%28v%2Bf%29=-%28u%2Bg%29%28x-u%29\"$
\n" ); document.write( " $\"%28y-v%29%28v%2Bf%29%2B%28u%2Bg%29%28x-u%29=0\"$
\n" ); document.write( " $\"vy%2Bfy-v%5E2-fv%2Bux-u%5E2%2Bgx-gu=0\"$
\n" ); document.write( " $\"%28u%2Bg%29x%2B%28v%2Bf%29y-%28u%5E2%2Bv%5E2%2Bgu%2Bfv%29=0\"$
\n" ); document.write( "Comparing,
\n" ); document.write( " $\"l=u%2Bg\"$
\n" ); document.write( " $\"m=v%2Bf\"$
\n" ); document.write( " $\"n=-%28u%5E2%2Bv%5E2%2Bgu%2Bfv%29\"$ \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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