document.write( "Question 1091873: Find 2 equation of a circle that touch the line y=-x+9, the positive y-axis, and positive x-axis \n" ); document.write( "

Algebra.Com's Answer #706395 by ikleyn(52794) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Find 2 $\"highlight%28cross%28equation%29%29\"$ equations of a circle that touch the line y=-x+9, the positive y-axis, and positive x-axis.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "1.  Since the circle touches the positive y-axis and positive x-axis,  its center lies in the angle bisector of the angle\r\n" );
document.write( "    concluded between the axes x and y.\r\n" );
document.write( "\r\n" );
document.write( "    In other words, the center lies in the straight line y = x.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2.  There are two such circles in the Quadrant I.\r\n" );
document.write( "\r\n" );
document.write( "    The smaller circle lies inside the right angled triangle formed by the coordinate x- and y-axes and the straight line y = -x + 9. \r\n" );
document.write( "\r\n" );
document.write( "    It is inscribed circle to this triangle.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    The other, the larger circle lies outside this triangle and makes external touching with its hypotenuse.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "3.  Let us solve the problem for the larger circle first.\r\n" );
document.write( "\r\n" );
document.write( "    From what I said above, it is clear that the point (4.5,4.5) lies in the circle and is the touching point.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Now, if you make a sketch, you will easily get the equation for the radius R of the circle:\r\n" );
document.write( "\r\n" );
document.write( "         = ,\r\n" );
document.write( "\r\n" );
document.write( "    which implies\r\n" );
document.write( "\r\n" );
document.write( "         =   ====>  . =   ====>  R =  =  = .\r\n" );
document.write( "\r\n" );
document.write( "    Thus the coordinates of the center are (x,y) = (R,R) and the radius is R =  = 15.364    (approximately)    (1)\r\n" );
document.write( "\r\n" );
document.write( "    The equation of the larger circle is   +  =   with  R  expressed in the formula (1).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "4.  Let us solve the problem for the smaller circle now.\r\n" );
document.write( "\r\n" );
document.write( "    If you make a sketch, you will easily get the equation for the radius r of the smaller circle:\r\n" );
document.write( "\r\n" );
document.write( "         = ,\r\n" );
document.write( "\r\n" );
document.write( "    which implies\r\n" );
document.write( "\r\n" );
document.write( "         =   ====>  . =   ====>  r =  =  = .\r\n" );
document.write( "\r\n" );
document.write( "    Thus the coordinates of the center are (x,y) = (r,r) and the radius is r =  = 2.636   (approximately)    (2)\r\n" );
document.write( "\r\n" );
document.write( "    The equation of the smaller circle is   +  =   with  r  expressed by the formula (2).\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );