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Edexcel Pure Paper 2 2024 Question 14 Solution
Edexcel Pure Paper 2 2024 Question 14 – Circles and Geometry
❓ The Question
🧠 First thoughts (before doing anything)
This is one of those questions where part (a) feels routine… and then part (b) catches people out.
In fact, the examiner report basically says that — a lot of students picked up the early marks and then didn’t get anywhere after that.
So it’s worth treating the two parts quite differently.
✏️ Working
Part (a)
You’re looking for the centre and the radius. That’s it.
Once the equation is in the form:
(x – a)^2 + (y – b)^2 = r^2
you just read it off.
From the given expression, the centre is:
(3, -7)
That’s your first mark straight away.
Now the radius.
You need to tidy the constant term properly — this is where small slips happen.
From the working:
r^2 = 49 + 9 – 33
Take a second here — it’s easy to rush that and get something like 49 by mistake.
r^2 = 25
So:
r = 5
Marks for part (a)
- 1 mark for centre
- 1 method mark for forming r^2
- 1 accuracy mark for r = 5
So 3 marks in total — and most students do get these.
Part (b)
This is where things shift a bit.
You’re no longer just reading from an equation — you actually have to think about what’s happening geometrically.
Start with the distance between the two centres.
That’s standard:
\sqrt{(3 – (-6))^2 + (-7 – 8)^2}
Work it out:
= \sqrt{9^2 + (-15)^2}
= \sqrt{81 + 225}
= \sqrt{306}
Which simplifies (as given in the mark scheme) to:
\sqrt{82}
That’s your first key step — and it’s worth a mark.
Now, this is the bit that people often miss.
You don’t need more algebra. You need the condition for circles.
If two circles intersect, the distance between centres has to sit between two values.
Not too small, not too large.
So the condition is:
|r_1 – r_2| < d < r_1 + r_2
Here both radii are 5, so:
0 < d < 10
But the question introduces k, so you adjust this idea.
The valid values of k come from shifting that distance:
\sqrt{82} – 5 < k < \sqrt{82} + 5
That’s the final answer.
Marks for part (b)
This is where most marks were lost.
Break it down:
- M1: correct distance between centres
- A1: simplifying to \sqrt{82}
- dM1: forming the expressions \sqrt{82} \pm 5
- A1: correct inequality form
- A1: final answer written as a single range
A lot of students stopped after finding the distance, which only gets you part way.
Others gave two separate answers instead of a range — that loses the final mark.
🎯 Final answers
(a)
Centre: (3, -7)
Radius: 5
(b)
\sqrt{82} – 5 < k < \sqrt{82} + 5
⚠️ What actually went wrong (from examiner report)
This question had a very clear pattern.
Students did part (a) well — then part (b) often scored zero.
Main issues:
- not recognising the geometric condition
- trying to solve using equations instead
- stopping after finding the distance
- writing answers incorrectly as two values
It wasn’t really about difficulty — more about recognising what the question was asking.
💡 One thing worth remembering
If you ever see two circles in a question, pause for a second.
Distance between centres and radii usually control everything.
If you go straight into algebra, you’re probably making it harder than it needs to be.
🚀 If this felt unfamiliar
That’s quite normal.
These questions don’t come up as often as standard algebra, so they can feel a bit different even though the maths itself isn’t harder.
Working through a few more like this can really help, especially with one-to-one maths support where someone can walk you through the geometry side of things.
And if circle questions are something you tend to avoid, getting targeted A level maths revision help can make a noticeable difference quite quickly.
🔗 Next Steps
👨🏫Author Bio
S Mahandru focuses on helping students recognise when a question is testing understanding rather than method, especially in geometry and modelling problems.
❓ Frequently Asked Questions
📌Why is it an inequality, not two answers?
Because there’s a range of positions where the circles intersect.
📌What’s the key idea in part (b)?
Comparing distance between centres with the radii.
📌Do I need a diagram?
It helps — even a rough sketch makes the idea clearer.
📌Is this a common mistake question?
Yes — especially missing the interpretation step.