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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:
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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:
The perpendicular distance $d$ between the point $(x_0,y_0)$ and the line $y=mx+b$ is given by:
$\displaystyle d=\frac{|mx_0+b-y_0|}{\sqrt{m^2+1}}$
$\displaystyle d=\frac{|mx_0+b-y_0|}{\sqrt{m^2+1}}$
--------------------
Remember to read the POTW submission guidelines to find out how to submit your answers!
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