A2 June 2019 Paper 2 Q6

AQACurrent spec6 marksComplex Numbers

6 A circle \(C\) in the complex plane has equation \(|z - 2 - 5\mathrm{i}| = a\)

The point \(z_1\) on \(C\) has the least argument of any point on \(C\), and \(\arg(z_1) = \dfrac{\pi}{4}\)

Prove that \(a = \dfrac{3\sqrt{2}}{2}\) [6 marks]

Scheme	Marks	AO
Sets as equal the real and imaginary parts of \(z_1\) or uses the line \(y = x\)	M1	2.2a
Uses the fact that the half-line is a tangent to the circle	M1	2.2a
Forms an equation for the gradient of the line joining (2, 5) and \((k, k)\) or the gradient of a tangent at any point on \(C\) or uses a right angled triangle containing the line joining (2, 5) and \((k, k)\) or correctly substitutes \(y = x\) into the equation of the circle	M1	3.1a
Forms a correct equation based on their method. PI by \(k = 3.5\)	M1	2.2a
Obtains the value of \(a\) from their equation	A1	1.1b
Produces a completely correct, rigorous proof leading to exact value of \(a\). Must show all steps clearly	R1	2.1
	(6 marks)