The origin O and a point B(p, q) are opposite vertices of the square OABC. Find the coordinates of the point A and C

A line l has gradient q/p. Find possible values for the gradient of a line at 45° to l.

So I’m stuck on the first part of this problem and haven’t tried to tackle the second part yet, but I posted it anyway in case I have difficulties with it once I have solved the first part.

For the first question I observed that I have 4 unknowns, the two coordinates of each point. So I figured I should determine 4 equations that would help me find these; I used the following remarks to define these equations:

The gradient of AC will be perpendicular to the gradient of OB,

The gradient of OA will be perpendicular to the gradient of AB

The gradient of OC will be perpendicular to the gradient of CB,

The distance between A and C will be equal to the distance between O and B

However I’m not sure that this is the right (or at least, most effective) way to proceed as I’ve ended up with a mess of equations, often ending up quadratic, that appear to be leading me nowhere. Am I on the right track, have I used the wrong equations or am I just using a totally incorrect method?

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