A complementary-angle identity and a shifted-curve area

1. Using a right-angled triangle, show that, for \(a>0\), \[\tan^{-1}(a)+\tan^{-1}\left(\frac1a\right)=\frac{\pi}{2}.\][2]

The curves \(D_1\) and \(D_2\) have equations \[D_1:\ y=\frac{2}{1+(x-1)^2},\qquad D_2:\ y=\frac{k}{12x-7},\] where \(k>0\).

2. Determine the values of \(k\) for which the curves have at least one point of intersection.[3]

For the remaining parts, take \(k=5\).

3. Sketch both curves on the same axes. Label their intersections with the coordinate axes, the maximum point of \(D_1\), and all asymptotes.[3]
4. Calculate the exact area of the finite region enclosed by the curves, simplifying your answer using part (i).[5]