# Hyperbola Problem

• Sep 12th 2008, 01:50 PM
jeta
Hyperbola Problem
I'm having trouble with this problem. Any help would be appreciated.

The normal at any point on the hyperbola x^2/a^2 - y^2/b^2 = 1 intersects the x and y axes at P and Q respectively. Find the locus of the midpoint of PQ.

Thanks
• Sep 12th 2008, 06:49 PM
mr fantastic
Quote:

Originally Posted by jeta
I'm having trouble with this problem. Any help would be appreciated.

The normal at any point on the hyperbola x^2/a^2 - y^2/b^2 = 1 intersects the x and y axes at P and Q respectively. Find the locus of the midpoint of PQ.

Thanks

The hyperbola can be defined by the following parametric equations:

$\displaystyle x = a \sec t = \frac{a}{\cos t}$

$\displaystyle y = b \tan t = \frac{b \sin t}{\cos t}$.

Then $\displaystyle \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} = -\frac{b}{a} \frac{1}{\sin t}$.

The equation of the tangent can be written using the model $\displaystyle y - y_1 = m (x - x_1)$: $\displaystyle y - \frac{b \sin t}{\cos t} = -\frac{b}{a} \frac{1}{\sin t} \left( x - \frac{a}{\cos t} \right)$.

Use this equation to get the coordinates of the x- and y-intercepts, that is, the coordinates of P and Q. Use these coordinates to get the midpoint of PQ.

You now have the parametric equations of the required locus. Convert them into a Cartesian equation if required.

Warning: My carelessness is a well documented fact. The above results may contain errors. You should work through the calculations very carefully for yourself.
• Sep 12th 2008, 07:42 PM
jeta
Thanks alot for your help.(Happy)