Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 7 / a / i

Table of contents
- Solution i

Solution

Write

z = x + i y

and square the modulus equation. It becomes

x^{2} + (y + b)^{2} = P^{2} [x^{2} + (y - b)^{2}] .

For

P \neq = 1

, completing the square gives the Apollonius circle

x^{2} + (y - \frac{b ( P ^{2} + 1 )}{P ^{2} - 1})^{2} = \frac{4 b ^{2} P ^{2}}{( P ^{2} - 1 ) ^{2}} .

Hence

center (0, \frac{b ( P ^{2} + 1 )}{P ^{2} - 1}), radius \frac{2 b P}{∣ P ^{2} - 1∣} .

The printed statement needs the exception

P = 1 ⟹ y = 0

: equal distances from

ib

and

- ib

give their perpendicular bisector, a straight line rather than a finite-radius circle. For

P > 1

the center lies in the upper half-plane; this is the case needed in part (ii). The original PDF confirms the numerator

z + ib

, correcting the duplicated, corrupted OCR formula.

 Ancestors (13)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version