document.write( "Question 983166: please help me to solve this problem:
\n" ); document.write( "Find the equation of the tangent to the hyperbola x^2-4y^2=36 which is perpendicular to the line x-y+4=0
\n" ); document.write( " Thanks a lot \n" ); document.write( "

Algebra.Com's Answer #604204 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
A line perpendicular to $\"x-y%2B4=0\"$ would have a slope that is the negative reciprocal of its slope.
\n" ); document.write( " $\"y=x%2B4\"$
\n" ); document.write( " $\"m=1\"$
\n" ); document.write( " $\"m%5Bp%5D=-1\"$
\n" ); document.write( "So the slope of the perpendicular line would be $\"m=-1\"$ .
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\n" ); document.write( "The tangent line to the hyperbola can be found by differentiating,
\n" ); document.write( " $\"2x-8y=0\"$
\n" ); document.write( " $\"2xdx=8ydy\"$
\n" ); document.write( " $\"m=dy%2Fdx=%282x%29%2F%288y%29=x%2F%284y%29\"$
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\n" ); document.write( "This slope must equal the previously calculated slope,
\n" ); document.write( " $\"x%2F%284y%29=-1\"$
\n" ); document.write( " $\"4y=-x\"$
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\n" ); document.write( " $\"x%5E2%281-1%2F4%29=36\"$
\n" ); document.write( " $\"%283%2F4%29x%5E2=36\"$
\n" ); document.write( " $\"x%5E2=48\"$
\n" ); document.write( " $\"x=0+%2B-+4sqrt%283%29\"$
\n" ); document.write( "So then,
\n" ); document.write( " $\"y=0+%2B-+sqrt%283%29\"$
\n" ); document.write( "The two points of intersection are,
\n" ); document.write( "( $\"-4sqrt%283%29\"$ , $\"sqrt%283%29\"$ ) and ( $\"4sqrt%283%29\"$ , $\"-sqrt%283%29\"$ )
\n" ); document.write( "So then using the point-slope form of a line,
\n" ); document.write( " $\"y-sqrt%283%29=-%28x%2B4sqrt%283%29%29\"$ and $\"y%2Bsqrt%283%29=-%28x-4sqrt%283%29%29\"$
\n" ); document.write( " $\"highlight_green%28y=-x%2B3sqrt%283%29%29\"$ and $\"highlight_green%28y=-x-3sqrt%283%29%29\"$
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.\r
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