Standard +0.8 This requires converting a polar equation to Cartesian form to identify the circle's radius, involving algebraic manipulation and completing the square. While the technique is standard for Further Maths students, the non-trivial polar form and need for full justification elevate it above routine exercises, placing it moderately above average difficulty.
The polar equation of the circle \(C\) is
$$r = a(\cos \theta + \sin \theta)$$
Find, in terms of \(a\), the radius of \(C\).
Fully justify your answer.
[4 marks]
states that the circle must pass through , and that the
maximum value of is 1.
ππ ππ
Answer
Marks
Guidance
4
1.1a
M1
cosοΏ½ππβ οΏ½
Obtains the correct radius = .
Answer
Marks
Guidance
ππ
3.2a
A1
β2
Answer
Marks
Guidance
Total
4
Q
Marking instructions
AO
Question 17:
17 | Selects a method to transform the given equation of C
into a standard polar form or Cartesian form, e.g.
uses and and ;
or 2 2 2
writes ππ in= tπ₯π₯he +foπ¦π¦rm π₯π₯ = ππcosππ π¦π¦ = ππ. sinππ | 3.1a | M1 | 2
ππ = ππ(ππcosππ+ππsin ππ)
2 2
π₯π₯ +π¦π¦ = ππ(π₯π₯+π¦π¦)
2 2
π₯π₯ βπππ₯π₯+π¦π¦ βπππ¦π¦ = 0
2 2 2
2 ππ 2 ππ ππ
π₯π₯ βπππ₯π₯+ +π¦π¦ βπππ¦π¦+ =
4 4 2
2 2 2
ππ ππ ππ
οΏ½π₯π₯β2οΏ½ +οΏ½π¦π¦β2οΏ½ = οΏ½β2οΏ½
radius
ππ
= β2
ππ π π (cosπ΄π΄cosπ΅π΅+sinπ΄π΄sinπ΅π΅)
Obtains a correct equation in terms of and only
or
obtains π₯π₯ π¦π¦
.
ππ
4 | 1.1b | A1
ππ = ππβ2cosοΏ½ππβ οΏ½
Correctly completes the square of their quadratic
expression
or
states that the circle must pass through , and that the
maximum value of is 1.
ππ ππ
4 | 1.1a | M1
cosοΏ½ππβ οΏ½
Obtains the correct radius = .
ππ | 3.2a | A1
β2
Total | 4
Q | Marking instructions | AO | Marks | Typical solution
The polar equation of the circle $C$ is
$$r = a(\cos \theta + \sin \theta)$$
Find, in terms of $a$, the radius of $C$.
Fully justify your answer.
[4 marks]
\hfill \mbox{\textit{AQA Further AS Paper 1 2020 Q17 [4]}}