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- Jan 26, 2012

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Thank you to MarkFL for this week's problem!

For the hyperbolas:

$\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=\pm1$

Demonstrate that the product of the perpendicular distances from a arbitrary point on either hyperbola to its asymptotes is constant, and give the value of this constant as a function of the parameters.

Hint:

The perpendicular distance $d$ between the point $(x_0,y_0)$ and the line $y=mx+b$ is given by:

$\displaystyle d=\frac{|mx_0+b-y_0|}{\sqrt{m^2+1}}$

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Remember to read the POTW submission guidelines to find out how to submit your answers!

For the hyperbolas:

$\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=\pm1$

Demonstrate that the product of the perpendicular distances from a arbitrary point on either hyperbola to its asymptotes is constant, and give the value of this constant as a function of the parameters.

Hint:

$\displaystyle d=\frac{|mx_0+b-y_0|}{\sqrt{m^2+1}}$

--------------------

Remember to read the POTW submission guidelines to find out how to submit your answers!

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