document.write( "Question 1194922: the tangent and normal to the curve y=4*sqrt(x+2) at the point P(7,12) cut the x-axis at M and N respectively. Calculate the area of the triangle PMN \n" ); document.write( "

Algebra.Com's Answer #827226 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( " $\"y=4sqrt%28x%2B2%29=4%28x%2B2%29%5E%281%2F2%29\"$

\n" ); document.write( " $\"dy%2Fdt=%281%2F2%29%284%29%28x%2B2%29%5E%28-1%2F2%29=2%2Fsqrt%28x%2B2%29\"$

\n" ); document.write( "At (7,12), the slope of the tangent line is $\"2%2Fsqrt%289%29=2%2F3\"$ , so the slope of the normal line is -3/2.

\n" ); document.write( "Find the x-intercepts of the tangent line and normal line by setting y=0:

\n" ); document.write( " $\"y-12=%282%2F3%29%28x-7%29\"$
\n" ); document.write( " $\"-12=%282%2F3%29%28x-7%29\"$
\n" ); document.write( " $\"-18=x-7\"$
\n" ); document.write( " $\"x=-11\"$

\n" ); document.write( " $\"y-12=%28-3%2F2%29%28x-7%29\"$
\n" ); document.write( " $\"8=x-7\"$
\n" ); document.write( " $\"x=15\"$

\n" ); document.write( "The two x-intercepts are -11 and 15, so the triangle has base length 15-(-11)=26 and height 12; its area is (1/2)(26)(12)=156

\n" ); document.write( "ANSWER: 156

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