Sketch the locus of z on an Argand diagram if arg[(z-5)/(z-3)] = π/6

Rewrite as arg(z-5) - arg(z-3) = π/6 and let arg(z-5) = b and arg(z-3) = a, so that b-a = π/6Since we know that each argument makes a half line (starting at (3,0) for angle a, (5,0) for angle b) the half lines must intersect at a point P which is on the locus of z. The angle formed by this intersection must be equal to b-a = π/6 since the exterior angle in a triangle (in this case b) is equal to the sum of the interior angles (in this case a and π/6).We know from circle theorems that the angles subtended at the circumference in the same segment are always equal. Hence we can deduce that since the angle formed by the intersection is constant (equal to π/6) as b and a both vary, the locus of z must be an arc of a circle from x=3 to x=5 for y>0 (since the angle is positive).

FA
Answered by Faris A. Further Mathematics tutor

4963 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

Solve this equation: x^2 + 2x + 2


How do you plot a complex number in an Argand diagram?


A rectangular hyperbola has parametric equations x = 4t, y = 4/t , (z non 0). Points P and Q on this hyperbola have parameters t = 1/4 and t = 2. Find the equation of the line l which passes through the origin and is perpendicular to the line PQ.


State the conditions by which a Poisson distribution model may be suitable for a given random variable X.


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences