HARD Hyperbola intersecting ellipse question. find acute angle between tangents!!!

**differentiate** · January 10th 2010, 04:15 PM

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$ has an eccentricity of e. The ellipse $\frac{x^2}{a^2+b^2}+\frac{y^2}{b^2} = 1$ has an eccentricity of 1/e.
If the two graphs intersect at P in the first quadrant, show that the acute angle $\Theta$ between the the tangents to the curves at P satisfies:

$tan\Theta = \sqrt{2}(e + \frac{1}{e})$

Thanks!

**HallsofIvy** · January 11th 2010, 04:43 AM

Use implicit differentiation to find the slope of the tangent line of each graph at point (x,y). The slope is, of course, the tangent of the angle each makes with the x-axis. And the angle between the tangent lines (which is, by definition, the angle between the curves) is the difference between those two angles. Use the trig indentity $tan(x- y)= \frac{tan(x)- tan(y)}{1+ tan(x)tan(y)}$ .

You will need to know, if you do not already, that the eccentricity of the ellipse, $\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1$ , is given by $e= \frac{\sqrt{a^2- b^2}}{a}$ (for a> b) and that the eccentricity of the hyperbola, $\frac{x^2}{a^2}- \frac{y^2}{b^2}= 1$ , is given by $e= \frac{\sqrt{a^2+ b^2}}{a}$ (also for a> b).

**Jhevon** · January 11th 2010, 05:58 PM

Originally Posted by HallsofIvy

You will need to know, if you do not already, that the eccentricity of the ellipse, $\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1$ , is given by $e= \sqrt{|a^2- b^2|}$ and that the eccentricity of the hyperbola, $\frac{x^2}{a^2}- \frac{y^2}{b^2}= 1$ , is given by $e= \sqrt{a^2+ b^2}$ .

for your eccentricities: shouldn't those be divided by a?

**HallsofIvy** · January 12th 2010, 05:19 AM

Originally Posted by Jhevon

for your eccentricities: shouldn't those be divided by a?

Yes, thanks. I have gone back and corrected my post. What I gave was the "linear eccentricity".

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