document.write( "Question 989033: If y= mx + c is a tangents to the circle x^2 + y^2 =r^2,show that c = rsqrt(1 + m^2}}}. Hence, find the equations of the tangents to the circle x^2 + y^2 = 4 which pass through the points (0,+_6). \n" ); document.write( "

Algebra.Com's Answer #613198 by anand429(138) $\"\"$ $\"About$

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Centre of $\"x%5E2+%2B+y%5E2+=r%5E2\"$ is (0,0) and radius r.
\n" ); document.write( "Since y=mx+c (or say mx-y+c=0) is tangent to this circle, distance from centre(0,0) to this line is equal to radius
\n" ); document.write( "So,
\n" ); document.write( " $\"%28m%2A0-0%2Bc%29%2Fsqrt%28m%5E2%2B%28-1%29%5E2%29+=+r\"$
\n" ); document.write( "=> $\"c%2Fsqrt%28m%5E2%2B1%29+=+r\"$
\n" ); document.write( "=> $\"c+=+r%2Asqrt%28m%5E2%2B1%29\"$ --------------part (i)\r
\n" ); document.write( "\n" ); document.write( "Let y=mx+c be tangents to circle x^2 + y^2 = 4
\n" ); document.write( "Since, it passes through (0,6) and (0,-6)
\n" ); document.write( "So,
\n" ); document.write( "6=0+c and -6 = 0+c
\n" ); document.write( "=> c=6 or -6
\n" ); document.write( "Now, using part(i) proof,
\n" ); document.write( " $\"6=2%2Asqrt%28m%5E2%2B1%29\"$ or $\"-6=2%2Asqrt%28m%5E2%2B1%29\"$
\n" ); document.write( "=> $\"m=2sqrt%282%29\"$ or $\"m=-2sqrt%282%29\"$ (from both equations-same values of m)
\n" ); document.write( "So equation of tangents are
\n" ); document.write( " $\"y=2sqrt%282%29x%2B6\"$ and $\"y=2sqrt%282%29x-6\"$ and $\"y=-2sqrt%282%29x%2B6\"$ and $\"y=-2sqrt%282%29x-6\"$ \r
\n" ); document.write( "\n" ); document.write( "Since there are two external points, hence two tangents can be drawn from each point. So, we have got 4 equations of tangents. \n" ); document.write( "

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