SOLUTION: A curve has the equation {{{y = (ax+3)ln(x)}}}, where x is greater than zero and a is a constant. The normal to the curve at the point where the curve crosses the x-axis is paralle

Algebra ->  Coordinate-system -> SOLUTION: A curve has the equation {{{y = (ax+3)ln(x)}}}, where x is greater than zero and a is a constant. The normal to the curve at the point where the curve crosses the x-axis is paralle      Log On


   

Question 470860: A curve has the equation y+=+%28ax%2B3%29ln%28x%29, where x is greater than zero and a is a constant. The normal to the curve at the point where the curve crosses the x-axis is parallel to the line 5y+%2B+x+=+2. Find the value of a.

*Please answer as soon as possible bro : =)
Answer by Tatiana_Stebko(1539) About Me  (Show Source):
You can put this solution on YOUR website!
if the equation of the curve is y = f(x) then the slope of the normal line is m=-1/(f'(x0))
f'(x)=((ax+3)lnx)'=(ax+3)'lnx+(ax+3)(lnx)'=a*lnx+(ax+3)/x
the point where the curve crosses the x-axis is (x0,0)
find xo
Put into the curve equation y=0
%28ax%2B3%29lnx=0
x=1
then m=f'(x0)=f'(1)=a%2Aln1%2B%28a%2B3%29%2F1=a%2B3
the slope of the normal line to the curve at the point where the curve crosses the x-axis is -1/(f'(1))=-1%2F%28a%2B3%29
If normal is parallel to the line 5y+%2B+x+=+2, then their slopes are equal
5y+%2B+x+=+2=> 5y=-x%2B2=> y=%28-1%2F5%29x%2B2%2F5=> the slope m=-1%2F5
-1%2F%28a%2B3%29=-1%2F5
a%2B3=5
a=2