document.write( "Question 141508: circle with equation: X^2 + Y^2 - 6X - 18Y + 45
\n" ); document.write( "Tangent touches the circle at point T equation of the tangent is 2Y = X
\n" ); document.write( "Show algebraically that T has coordinates (6,3).
\n" ); document.write( "

Algebra.Com's Answer #103115 by Edwin McCravy(20086)\"\" \"About 
You can put this solution on YOUR website!
circle with equation: \"X%5E2+%2B+Y%5E2+-+6X+-+18Y+%2B+45=0\"
\n" ); document.write( "Tangent touches the circle at point T equation of the tangent is \"2Y+=+X\"
\n" ); document.write( "Show algebraically that T has coordinates (6,3).
\n" ); document.write( "
\r\n" );
document.write( "Just solve the system\r\n" );
document.write( "\r\n" );
document.write( "\"X%5E2+%2B+Y%5E2+-+6X+-+18Y+%2B+45+=0\"\r\n" );
document.write( "\"2Y+=+X\"\r\n" );
document.write( "\r\n" );
document.write( "by substitution\r\n" );
document.write( "\r\n" );
document.write( "Substitute 2Y for X in the first:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"X%5E2+%2B+Y%5E2+-+6X+-+18Y+%2B+45+=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"%282Y%29%5E2+%2B+Y%5E2+-+6%282Y%29+-+18Y+%2B+45+=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"4Y%5E2%2BY%5E2-12Y-18Y%2B45=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"+5Y%5E2-30Y%2B45=0+\"\r\n" );
document.write( "\r\n" );
document.write( "Divide every term by 5\r\n" );
document.write( "\r\n" );
document.write( "\"Y%5E2-6Y%2B9=0\"\r\n" );
document.write( "\r\n" );
document.write( "Factor:\r\n" );
document.write( "\r\n" );
document.write( "(Y-3)(Y-3)=0\r\n" );
document.write( "\r\n" );
document.write( "\"Y-3=0\" gives \"Y=3\"\r\n" );
document.write( "\"Y-3=0\" gives \"Y=3\"\r\n" );
document.write( "\r\n" );
document.write( "[The fact that we get only one solution tells us\r\n" );
document.write( "that the line touches the circle in only one\r\n" );
document.write( "point. So this proves that it is indeed tangent]\r\n" );
document.write( "\r\n" );
document.write( "Now to find \"X\"\r\n" );
document.write( "\r\n" );
document.write( "Substitute \"3\" for \"Y\" in\r\n" );
document.write( "\r\n" );
document.write( "\"2Y+=+X\"\r\n" );
document.write( "\r\n" );
document.write( "\"2%283%29=X\"\r\n" );
document.write( "6=X\r\n" );
document.write( "\r\n" );
document.write( "So the point of tangency is (6,3)\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );