Question
The equation of a line \(L_1\) is \(2 x+3 y=-5\).
(a) Find the gradient of \(L_1\).
A second line, \(L_2\), is perpendicular to \(L_1\).
(b) State the gradient of \(L_2\).
The point \(\mathrm{P}(4,0)\) lies on \(L_2\).
(c) Find the equation of \(L_2\), giving your answer in the form \(a x+b y+d=0\), where \(a, b, d \in \mathbb{Z}\).
The point \(\mathrm{Q}\) is the intersection of \(L_1\) and \(L_2\),
(d) Find the coordinates of Q.
Question
Consider the curve \(y=(k x-1) \ln (2 x)\) where \(k \in \mathbb{R}\) and \(x>0\).
The tangent to the curve at \(x=2\) is perpendicular to the line \(y=\frac{2}{5+4 \ln 4} x\).
Find the value of \(k\).