document.write( "Question 1139514: The circle is tangent to the line 4x+3y=4 at the point (4,-4) and center is on the line x-y=7. Find the equation of the circle. (I'm struggling to find the center in x-y=7 that says to be perpendicular to the center itself and to the point of tangency) \n" ); document.write( "

Algebra.Com's Answer #759980 by greenestamps(13209) $\"\"$ $\"About$

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\n" ); document.write( "Put the equations in slope-intercept form.

\n" ); document.write( " $\"4x%2B3y=4\"$ --> $\"y+=+-%284%2F3%29x%2B4%2F3\"$
\n" ); document.write( " $\"x-y=7\"$ --> $\"y+=+x-7\"$

\n" ); document.write( "If the circle is tangent to the line $\"y+=+-%284%2F3%29x%2B4%2F3\"$ at (4,-4), then the center of the circle has to lie on the line with slope 3/4 passing through (4,-4). (A radius to a point of tangency is perpendicular to the tangent; slopes of perpendicular lines are negative reciprocals.)

\n" ); document.write( " $\"y+=+%283%2F4%29x%2Bb\"$
\n" ); document.write( " $\"-4+=+%283%2F4%294%2Bb\"$
\n" ); document.write( " $\"-4+=+3%2Bb\"$
\n" ); document.write( " $\"b+=+-7\"$

\n" ); document.write( "So the center of the circle lies on the line $\"y+=+%283%2F4%29x-7\"$ .

\n" ); document.write( "The given information says that the center of the circle lies on the line $\"y+=+x-7\"$ .

\n" ); document.write( "So the center of the circle is at the one point that lies on both $\"y+=+%283%2F4%29x-7\"$ and $\"y+=+x-7\"$ .

\n" ); document.write( "Solve that pair of equations to find the intersection point; that is the center of the circle. \n" ); document.write( "

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