11 A point has Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\)
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- Express \(r\) in terms of \(x\) and \(y\)
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- Express \(x\) in terms of \(r\) and \(\theta\)
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- The curve \(C _ { 1 }\) has the polar equation
$$r ( 2 + \cos \theta ) = 1 \quad - \pi < \theta \leq \pi$$
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- Show that the Cartesian equation of \(C _ { 1 }\) can be written as
$$a y ^ { 2 } = ( 1 + b x ) ( 1 + x )$$
where \(a\) and \(b\) are integers to be determined.
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- (ii) The curve \(C _ { 2 }\) has the Cartesian equation
$$a x ^ { 2 } = ( 1 + b y ) ( 1 + y )$$
where \(a\) and \(b\) take the same values as in part (c)(i).
Describe fully a single transformation that maps the curve \(C _ { 1 }\) onto the curve \(C _ { 2 }\)