Hyperbola Problem

**jeta** · September 12th 2008, 01:50 PM

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

**mr fantastic** · September 12th 2008, 06:49 PM

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:

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

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

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

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

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.

**jeta** · September 12th 2008, 07:42 PM

Thanks alot for your help.

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