vertices of a square, coordinate geometry and differentiation

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?

Re: vertices of a square, coordinate geometry and differentiation

https://lh4.googleusercontent.com/-o...800/square.png

Note that the result of counterclockwise rotation by 90 degrees of a vector (p, q) is (-q, p), and the result of clockwise rotation by 90 degrees of (p, q) is (q, -p).

Re: vertices of a square, coordinate geometry and differentiation

Re: vertices of a square, coordinate geometry and differentiation

Quote:

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

For this you need .

Re: vertices of a square, coordinate geometry and differentiation

hi HallsofIvy

shouldn't it be 1-tan(theta)tan(fi) in the denominator?

Re: vertices of a square, coordinate geometry and differentiation

the problem is, i don't know how to formualte the euqation of a circle (if that even exists) and i'm learning about vectors in the chapter immediately after this one, so niether of the methods proposed above apply to my level. any other suggestions? this can't be too hard.

Re: vertices of a square, coordinate geometry and differentiation