Question

consider the curve y = 2 log x, where log is the natural logarithm. let l be the tangent to that curve which passes through the origin, let p be the point of contact of l and that curve, and let me be the straight line perpendicular to the tangent l at p. we are to find the equations of the straight lines l and m and the area s of the region bounded by the curve y = 2 logx, the straight line m, and the x-axis.

Answer 1

Hello,
Let's assume P=(a,2ln(a))

Slope of the tangent = 2ln(a)/a

f(x)=2ln(x)
==>f'(x)=2/x ==>f'(a)=2/a

slope=f'(a)
==>2 ln(a) /a =2/a ==>ln (a)=1 ==>a=e.

slope= 2/a=2/e


Equation of the tangent: y=2/e *x

P=(e,2)

Equation of  m:
slope= -1/(2/e)=-e/2
Equation: y-2=(x-e)*(-e/2)
y=-e/2 *x + e²/2 +2

Intersection of m  and the x-axis:
0=-e/2 *x + e²/2 +2 ==>x=e-4/e

Area= (e-4/e)*2/2 =e-4/e