1. ## 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

2. 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.

3. Thanks alot for your help.