Rectangular hyperbola, chord, origin

[Appleton]

Homework Statement

Prove that the chord joining the points P(cp, c/p) and Q(cq, c/q) on the rectangular hyperbola xy = c^2 has the equation

x + pqy = c(p + q)

The points P, Q, R are given on the rectangular hyperbola xy = c^2 . prove that

(a) if PQ and PR are equally inclined to the axes of coordinates, then QR passes through the origin O.

(b) if angle QPR is a right angle, then QR is perpendicular to the tangent at P

Homework Equations

The Attempt at a Solution

I can prove the equation of the chord joining the 2 points, but I am having difficulty with (a).

What does "inclined to the axes of coordinates" mean?

If I interpret 2 lines "inclined to the axes of coordinates" to mean to 2 parallel lines I am unable to envisage 2 such lines where R and Q are not coincident so I think my interpretation is wrong.

Also would the case where Q and R are coincident not reveal a counter example or does the wording of the question imply that the points are unique?

 

[Mark44]

Homework Statement

Prove that the chord joining the points P(cp, c/p) and Q(cq, c/q) on the rectangular hyperbola xy = c^2 has the equation

x + pqy = c(p + q)

The points P, Q, R are given on the rectangular hyperbola xy = c^2 . prove that

(a) if PQ and PR are equally inclined to the axes of coordinates, then QR passes through the origin O.

(b) if angle QPR is a right angle, then QR is perpendicular to the tangent at P

Does the problem include a graph or picture? From the given information, the point P and Q are on one branch of the hyperbola, and it would seem that R is a point on the other branch.

Homework Equations

The Attempt at a Solution

I can prove the equation of the chord joining the 2 points, but I am having difficulty with (a).

What does "inclined to the axes of coordinates" mean?

I'm not sure, but what I think it means is this: PR makes an angle with the y-axis and PQ makes an angle with the x-axis (or possibly the other way around -- what I've described is consistent with the picture I drew). The phrase "inclined to the axes of coordinates" means that the two angles I described are equal, but I'm not certain of that. In any case, this seems to me to be an odd way to describe things.

If I interpret 2 lines "inclined to the axes of coordinates" to mean to 2 parallel lines I am unable to envisage 2 such lines where R and Q are not coincident so I think my interpretation is wrong.

Also would the case where Q and R are coincident not reveal a counter example or does the wording of the question imply that the points are unique?

My reading of the problem is that Q and R are on different branches, otherwise the segment QR couldn't go through the origin.

 

[Appleton]

Thanks for your reply Mark44. Sorry for my delay. Is your interpretation of 2 lines "equally inclined to the axes of coordinates" equivalent to saying that the lines are perpendicular?

 

[Mark44]

Thanks for your reply Mark44. Sorry for my delay. Is your interpretation of 2 lines "equally inclined to the axes of coordinates" equivalent to saying that the lines are perpendicular?

That's not what I was thinking, but it could be true. I would have to see a picture with the angles labelled to make sure, and might be able to use ordinary geometry to prove this. Again, because of the wording of the problem, I'm not sure exactly what the author of the problem is saying.