3D vectors - circle crossing a line , given a midpoint question

The question is as follows:

Screenshot 2025-11-10 101650.png

I understand it all until question d)

I have found a similar question online and the mark scheme for it seems to say that we know the position vector for A, OA, which in our case will be, the position vector for B is unknown, the position vector for the point of intersection is known too:

9i - j - k = OA
ai + bj + ck = OB
4i + 14j -4k = OX (point of intersection) (this was worked out in part c)

What is then said is that X is the midpoint of AB, which I'm having a hard time visualising, even after drawing a sketch of the entire thing. We don't know where the centre of the circle is, just that it lies on line 2. If the centre of the circle was at the point of intersection then it does make sense, but it isn't necessarily, I guess I'm just struggling to picture it.

I get that the distance between point A and the centre of the circle will be the same as the distance between point B and the centre of the circle, I just don't know how to proceed with working it out.

Reply 1

needsomehelpt

Wait I think I realise now that regardless of where the centre of the circle is, the point of intersection has nothing to do with it, just the fact that the distance between the point of intersection and A, and the distance between the point of intersection and B will always be the same, and i see that in sketches now, regardless of how far along the line l₂ the circle;s centre is.

(edited 5 months ago)

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