3.13: Surd point on a curve and stationary point

0606/11/M/J/21 — Question 8 · 8 marks

Do not use a calculator in this question.

A curve has equation

y = \left(x - \dfrac{1}{x}\right)^{2}

. The

x

-coordinate of a point

A

on the curve is

\dfrac{3}{2} + \sqrt{3}

(a)

Show that the coordinates of

A

can be written in the form

\left(p + q\sqrt{3},\ r + s\sqrt{3}\right)

, where

p

q

r

and

s

are integers.

[5]

(b)

Find the

x

-coordinate of the stationary point on the curve, giving your answer in the form

a + b\sqrt{3}

, where

a

and

b

are rational numbers.

[3]