document.write( "Question 1088210: Find the equations of the lines described. Give your answers in the form y=mx+b. tangent lines to x²+y²-22x+12y+137=0 at points where x=15 \n" ); document.write( "

Algebra.Com's Answer #702460 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
The slope of the tangent line is equal to the value of the derivative at the point.
\n" ); document.write( "Complete the square to get to center-radius form of a circle,
\n" ); document.write( " $\"%28x%5E2-22x%29%2B%28y%5E2%2B12y%29=-137\"$
\n" ); document.write( " $\"%28x%5E2-22x%2B121%29%2B%28y%5E2%2B12y%2B36%29=-137%2B121%2B36\"$
\n" ); document.write( " $\"%28x-11%29%5E2%2B%28y%2B6%29%5E2=20\"$
\n" ); document.write( "So implicitly differentiating,
\n" ); document.write( " $\"2%28x-11%29dx%2B2%28y%2B6%29dy=0\"$
\n" ); document.write( " $\"%28y%2B6%29dy=%2811-x%29dx\"$
\n" ); document.write( " $\"dy%2Fdx=%2811-x%29%2F%286%2By%29\"$
\n" ); document.write( "So at $\"x=15\"$ , find the y value,
\n" ); document.write( " $\"15%5E2%2By%5E2-22%2815%29%2B12y%2B137=0\"$
\n" ); document.write( " $\"225%2By%5E2-330%2B12y%2B137=0\"$
\n" ); document.write( " $\"y%5E2%2B12y%2B32=0\"$
\n" ); document.write( " $\"%28y%2B4%29%28y%2B8%29=0\"$
\n" ); document.write( "So then,
\n" ); document.write( " $\"x=15\"$
\n" ); document.write( " $\"y=-4\"$
\n" ); document.write( " $\"m%5B1%5D=%2811-15%29%2F%286-4%29=-4%2F2=-2\"$
\n" ); document.write( "and
\n" ); document.write( " $\"x=15\"$
\n" ); document.write( " $\"y=-8\"$
\n" ); document.write( " $\"m%5B2%5D=%2811-15%29%2F%286-8%29=-4%2F-2=2\"$
\n" ); document.write( "So then using the point-slope form of the line,
\n" ); document.write( " $\"+y%2B4=-2%28x-15%29+\"$
\n" ); document.write( "and
\n" ); document.write( " $\"+y%2B8=2%28x-15%29+\"$
\n" ); document.write( "I leave it to you to convert to slope-intercept form.
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