# [SOLVED] Hyperbola 2

• Apr 16th 2010, 05:04 PM
UltraGirl
[SOLVED] Hyperbola 2
Hello

P is a variable point on xy=c^2. The tangent at P cuts the x axis at A and y axis at B. Q is the fourth vertex of OAQB. Show that the locus of Q is xy = 4c^2.

Ummm could someone please explain to me how on earth do we find Q. The question does not provide any information as to what OAQB is, so how exactly do you know what to do to find Q?

I've drawn the diagram and my guess is that Q's also on the hyperbola...but I don't know how to find it's point :(
• Apr 16th 2010, 07:30 PM
Debsta
Quote:

Originally Posted by UltraGirl
Hello

P is a variable point on xy=c^2. The tangent at P cuts the x axis at A and y axis at B. Q is the fourth vertex of OAQB. Show that the locus of Q is xy = 4c^2.

Ummm could someone please explain to me how on earth do we find Q. The question does not provide any information as to what OAQB is, so how exactly do you know what to do to find Q?

I've drawn the diagram and my guess is that Q's also on the hyperbola...but I don't know how to find it's point :(

It works out if you assume OAQB is a rectangle ( so Q has the same x-coordinate as A and the y-coord of B). Obviously something is missing from the question.
• Apr 17th 2010, 05:58 AM
Hello UltraGirl
Quote:

Originally Posted by UltraGirl
Hello

P is a variable point on xy=c^2. The tangent at P cuts the x axis at A and y axis at B. Q is the fourth vertex of OAQB. Show that the locus of Q is xy = 4c^2.

Ummm could someone please explain to me how on earth do we find Q. The question does not provide any information as to what OAQB is, so how exactly do you know what to do to find Q?

I've drawn the diagram and my guess is that Q's also on the hyperbola...but I don't know how to find it's point :(

As Debsta has said, if we assume that OAQB is a rectangle, then, from my answer to your first post, the coordinates of Q are
$\Big(2ct, \frac{2c}{t}\Big)$
So if Q is represented by the general point $(x,y)$, we have:
$x = 2ct, y = \frac{2c}{t}$
i.e.
$xy = 4c^2$