AS June 2023 Paper 1 Q11

11 A point has Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\) where \(r \geqslant 0\) and \(-\pi \lt \theta \leqslant \pi\)

(a) Express \(r\) in terms of \(x\) and \(y\) [1 mark]

(b) Express \(x\) in terms of \(r\) and \(\theta\) [1 mark]

(i) Show that the Cartesian equation of \(C_1\) can be written as\[ay^2 = (1 + bx)(1 + x)\]

where \(a\) and \(b\) are integers to be determined. [4 marks]

(ii) The curve \(C_2\) has the Cartesian equation\[ax^2 = (1 + by)(1 + y)\]

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 \(C_2\) [2 marks]

Scheme	Marks	AO
Writes a correct expression for \(r\)	B1	1.2
	(1)

Scheme	Marks	AO
Writes a correct expression for \(x\)	B1	1.2
	(1)

Mark scheme (c)

Scheme	Marks	AO
(i) Substitutes their (a) and (b) to form an equation in \(x\) and \(y\) only.	M1	3.1a
Correctly removes the square root from their equation. Must be an equation in terms of \(x\) and \(y\) only.	M1	1.1a
Obtains a correct equation without roots.	A1	1.1b
Obtains the correct equation in the required form.	A1	3.2a
	(4)
(ii) Identifies a reflection. Must also specify a line of reflection (which could be wrong for this mark). or Identifies a rotation about \((0, 0)\)	M1	1.1a
Fully describes a correct transformation. Accept \(90^\circ\) rotation about \((0, 0)\) If a rotation direction is included, it must be anticlockwise.	A1	1.1b
	(2)
	(8 marks)

Typical solution

(i)

\[\begin{aligned} & r(2 + \cos\theta) = 1 \\ \Rightarrow \ & 2r + r\cos\theta = 1 \\ \Rightarrow \ & 2\sqrt{x^2 + y^2} + x = 1 \\ \Rightarrow \ & 2\sqrt{x^2 + y^2} = 1 - x \\ \Rightarrow \ & 4(x^2 + y^2) = (1 - x)^2 \\ \Rightarrow \ & 4x^2 + 4y^2 = 1 - 2x + x^2 \\ \Rightarrow \ & 4y^2 = 1 - 2x - 3x^2 \\ \Rightarrow \ & 4y^2 = (1 - 3x)(1 + x)\end{aligned}\]

(ii)

\(y\) is replaced with \(x\), and vice versa

Reflection in \(y = x\)