7 The diagram shows a curve which starts from the point \(A\) with coordinates ( 0,2 ). The curve is such that, at every point \(P\) on the curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 } s$$ where \(s\) is the length of the \(\operatorname { arc } A P\).
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1. 1. Show that $$\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \sqrt { 4 + s ^ { 2 } }$$ (3 marks)
   2. Hence show that $$s = 2 \sinh \frac { x } { 2 }$$
   3. Hence find the cartesian equation of the curve.
2. Show that $$y ^ { 2 } = 4 + s ^ { 2 }$$