11.
The curves \(y = \mathrm { h } ( x )\) and \(y = \mathrm { h } ^ { - 1 } ( x )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(y = \mathrm { h } ( x )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(y = \mathrm { h } ^ { - 1 } ( x )\) crosses the \(x\)-axis at D and the \(y\)-axis at C .
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- Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
- Determine the coordinates of A, B, C and D.
- Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined.
- Points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D lie on a circle.
Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a , b\) and \(r ^ { 2 }\) are constants to be determined.