2024 HCI Promo Q5

Show that the distance \(L\) between any point \(\left( x,y \right)\) on the curve \(y={{\mathrm{e}}^{2x}}-3\) and the fixed point \(A\left( 2,-1 \right)\) satisfies the equation \({{L}^{2}}={{\left( x-2 \right)}^{2}}+{{\left( {{\mathrm{e}}^{2x}}-2 \right)}^{2}}\).[2]

Using differentiation, find the \(x\)-coordinate of the point \(P\) on the curve that is closest to the point \(A\), leaving your answer in \(4\) decimal places. You do not need to show that \(P\) is closest to \(A\).[2]

A variable point \(Q\) moves along the curve \(y={{\mathrm{e}}^{2x}}-3\) such that its \(x\)-coordinate is increasing at a rate of \(0.5\) units per second. Find the exact rate of change of \(L\) at the instant when \(Q\) is on the \(y\)-axis.[2]