document.write( "Question 54478: Given a circle: (x-4)^2 + (y+5)^2 = 5, find the equation of the line (in slope-intercept form) that is tangent to this circle at the point (3,-7). Hint: use the fact that a tangent line is perpendicular to the radius of the circle at the point where they meet. \n" ); document.write( "

Algebra.Com's Answer #36744 by josmiceli(19441) $\"\"$ $\"About$

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The problem tells you that (3,-7) is a point on the circle, but is it really?
\n" ); document.write( "Check this by filling in 3 for x and -7 for y
\n" ); document.write( "(x-4)^2 + (y+5)^2 = 5
\n" ); document.write( "(3-4)^2 + (-7+5)^2 = 5
\n" ); document.write( "(-1)^2 + (-2)^2 = 5
\n" ); document.write( "1 + 4 = 5
\n" ); document.write( "So (3,-7) really is on the circle
\n" ); document.write( "Where is the center of the circle?
\n" ); document.write( "The equation is in the form
\n" ); document.write( "(x - a)^2 + (y - b)^2 = r^2
\n" ); document.write( "where (a,b) is the center, and r is the length of the radius
\n" ); document.write( "So the center is (4,-5) and r^2 = 5, so r = $\"0+%2B+sqrt%285%29\"$
\n" ); document.write( "Now you have the endpoints of the line, so use
\n" ); document.write( "

\n" ); document.write( "where
\n" ); document.write( " $\"x%281%29+=+3\"$
\n" ); document.write( " $\"y%281%29+=+-7\"$
\n" ); document.write( "and
\n" ); document.write( " $\"x%282%29+=+4\"$
\n" ); document.write( " $\"y%282%29+=+-5\"$
\n" ); document.write( "The key is to get this equation in the form $\"y+=+mx+%2B+b\"$
\n" ); document.write( "then the slope of any line perpendicular to it will have the
\n" ); document.write( "slope $\"-%281%2Fm%29\"$
\n" ); document.write( " $\"%28y+-+%28-7%29%29+%2F+%28x+-+3%29+=+%28-5+-+%28-7%29%29+%2F+%284+-+3%29\"$
\n" ); document.write( " $\"%28y+-+%28-7%29%29+%2F+%28x+-+3%29+=+2+%2F+1\"$
\n" ); document.write( "multiply both sides by (x - 3)
\n" ); document.write( " $\"y+%2B+7+=+2x+-+6\"$
\n" ); document.write( " $\"y+=+2x+-+13\"$
\n" ); document.write( "The slope m = 2, so the slope of a line perpendicular to this line
\n" ); document.write( "will have slope $\"-1%2Fm+=+-1%2F2\"$
\n" ); document.write( "This is the slope of the tangent line at (3,-7), so you can write
\n" ); document.write( " $\"m+=+%28y+-+%28-7%29%29+%2F+%28x+-+3%29\"$
\n" ); document.write( " $\"-1%2F2+=+%28y+%2B+7%29+%2F+%28x+-+3%29\"$
\n" ); document.write( "multiply both sides by 2
\n" ); document.write( " $\"+-1+=+%282y+%2B+14%29+%2F+%28x+-+3%29\"$
\n" ); document.write( "multiply both sides by x-3
\n" ); document.write( " $\"-x+%2B+3+=+2y+%2B+14\"$
\n" ); document.write( " $\"2y+=+-x+-+11\"$
\n" ); document.write( " $\"y+=+%28-1%2F2%29x+-%2811%2F2%29\"$
\n" ); document.write( "Does this line go through (3,-7) ? Check it
\n" ); document.write( " $\"-7+=+%28-1%2F2%29%2A3+-+%2811%2F2%29\"$
\n" ); document.write( " $\"-7+=+-3%2F2+-+11%2F2\"$
\n" ); document.write( " $\"-7+=+-14%2F2\"$
\n" ); document.write( " $\"-7+=+-7\"$
\n" ); document.write( "OK \n" ); document.write( "

\n" );