Can You Solve This Very Very Tricky Oxford University Maths Question?

Sometimes I ask myself, you know, what’s the reason for existence. What’s the purpose of life? I don’t have a sophisticated answer. But I believe in living for love. To do what one can for the pursuit of love and relationships. To do it for love.

That being said, every article I write on Medium is one step closer to that.

Hopefully this maths puzzle from the University of Oxford will give you something to think about.

Once again, this is a good moment to pause the article and give the problem a go yourself. When you’re ready, keep reading for the solution. And if you come up with your own approach, feel free to share it in the comments — I’d love to see how you tackled it.

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Solution

One might be tempted to use the idea of discriminant and find individual pairs of ms and cs. However, notice the question is specifically asking for the possibilities for the line y = mx +c which satisfy the given conditions.

If we substitute y = mx + c into the two equations, we get

These are two circles with radius one, one centred at the origin and the other centred at (3,1). If we plot them on a graph, we see that

These are two non-overlapping circles. Now, think carefully about what the question really is asking.

In fact, this is equivalent to asking, given an arbitrary straight-line equation, how many ways can we manipulate m and c in y = mx + c such that the line touches the two circles. We want the line to touch to two circles because it states that m and c are values such that both equations have a repeated root.

If the line crosses the circle, that’s not a repeated root. That’s when the discriminant is greater than zero.

So, how many lines can we fit between the two circles such that the lines just touch both of them?

There are indeed four such lines!

Here’s a challenge for all of us. Can you find the equations of those four straight lines? (I haven’t done that, but it’d be great if some of you could try it out!)

Therefore, there are four sets of values of ms and cs that satisfy the equations.

The answer is (e).

And that’s our answer. How amazing :)

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