document.write( "Question 1016362: an ellipse is defined by $\"x%5E2%2F81+%2B+y%5E2%2F64+=+1\"$ . Find the equations of the lines tangent to this ellipse which make and angel of 45 degrees with the x-axis\r
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Algebra.Com's Answer #632755 by Fombitz(32388) $\"\"$ $\"About$

You can put this solution on YOUR website!
Differentiate,
\n" ); document.write( " $\"%282xdx%29%2F81%2B%282ydy%29%2F64=0\"$
\n" ); document.write( " $\"%28ydy%29%2F64=-%28xdx%29%2F81\"$
\n" ); document.write( " $\"dy%2Fdx=-%28x%2Fy%29%2864%2F81%29\"$
\n" ); document.write( "The slope of the tangent line is equal to the value of the derivative.
\n" ); document.write( "An angle of 45 degrees is equivalent to a slope of 1.
\n" ); document.write( " $\"1=-%28x%2Fy%29%2881%2F64%29\"$
\n" ); document.write( " $\"y=-%2864%2F81%29x\"$
\n" ); document.write( "This point also satisfies the ellipse equation,
\n" ); document.write( " $\"x%5E2%2F81%2B%281%2F64%29%2864%2F81%29%5E2x%5E2=1\"$
\n" ); document.write( " $\"%281%2F81%2B64%2F81%5E2%29x%5E2=1\"$
\n" ); document.write( " $\"%28145%2F6561%29x%5E2=1\"$
\n" ); document.write( " $\"x%5E2=6561%2F145\"$
\n" ); document.write( " $\"x=0+%2B-+81%2Fsqrt%28145%29\"$
\n" ); document.write( " $\"x=0+%2B-+%2881%2F145%29sqrt%28145%29\"$
\n" ); document.write( "Then plugging that into the ellipse equation you get,
\n" ); document.write( " $\"y=+0+%2B-+%2864%29%2Fsqrt%28145%29\"$
\n" ); document.write( " $\"y=0+%2B-+%2864%2F145%29sqrt%28145%29\"$
\n" ); document.write( "So the two points are,
\n" ); document.write( "( $\"%2881%2F145%29sqrt%28145%29\"$ , $\"-%2864%2F145%29sqrt%28145%29\"$ )
\n" ); document.write( "( $\"-%2881%2F145%29sqrt%28145%29\"$ , $\"%2864%2F145%29sqrt%28145%29\"$ )
\n" ); document.write( "That's when the slope is 1.
\n" ); document.write( "Similarly when the slope is -1.
\n" ); document.write( "( $\"%2881%2F145%29sqrt%28145%29\"$ , $\"%2864%2F145%29sqrt%28145%29\"$ )
\n" ); document.write( "( $\"-%2881%2F145%29sqrt%28145%29\"$ , $\"-%2864%2F145%29sqrt%28145%29\"$ )
\n" ); document.write( "Use the point slope form of a line to get the equation of the tangent line.
\n" ); document.write( " $\"y%2B64%2Fsqrt%28145%29=1%28x-81%2Fsqrt%28145%29%29\"$
\n" ); document.write( " $\"y=x-145%2Fsqrt%28145%29\"$
\n" ); document.write( " $\"y=x-sqrt%28145%29\"$
\n" ); document.write( "Similarly,
\n" ); document.write( " $\"y=x%2Bsqrt%28145%29\"$
\n" ); document.write( " $\"y=-x%2Bsqrt%28145%29\"$
\n" ); document.write( " $\"y=-x-sqrt%28145%29\"$
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