2023 ASRJC Promo Q9

State a sequence of three transformations which transform the graph with equation \({{x}^{2}}+{{y}^{2}}=1\) to the graph with equation \(\frac{{{(x+1)}^{2}}}{4}+\frac{{{y}^{2}}}{9}=1\).[3]

Sketch, on the same diagram, the graphs of \(y=\left| 2x-1 \right|\) and \(9{{\left( x+1 \right)}^{2}}+4{{y}^{2}}=36\) for \(y\ge 0\). State clearly the axial intercepts of both the graphs and the coordinates of the intersections between the two graphs.[5]

Hence solve the inequality \(\sqrt{36-9{{\left( x+1 \right)}^{2}}}\le 2\left| 2x-1 \right|\).[2]

Using your answer in (iii), find the exact range of values of \(x\) for which \(\sqrt{36-9{{\left( \frac{1}{x}+1 \right)}^{2}}}\le 2\left| \frac{2}{x}-1 \right|\).[2]