2.30: Curve from normal gradient and tangent equation

0606/21/O/N/20 — Question 10 · 8 marks

The gradient of the normal to a curve at the point (x,y)(x, y) is given by xx+1\dfrac{x}{x + 1}.
(a)
Given that the curve passes through the point (1,4)(1, 4), show that its equation is y=5−ln⁡x−xy = 5 - \ln x - x.
[5]
(b)
Find, in the form y=mx+cy = mx + c, the equation of the tangent to the curve at the point where x=3x = 3.
[3]