DS' Random Work: Hyperbolas: Standard equation derivation

$\left( \sqrt{\left( x-c\right) ^{2}+y^{2}}+\sqrt{\left( x+c\right) ^{2}+y^{2}}\right) ^{2}=2a$

$\left( \sqrt{c^{2}-2cx+x^{2}+y^{2}}\sqrt{c^{2}+2cx+x^{2}+y^{2}}\right) ^{2}=2a^{2}-\left( c^{2}+x^{2}+y^{2}\right)$

$c^{4}-2c^{2}x^{2}+2c^{2}y^{2}+x^{4}+2x^{2}y^{2}+y^{4}=\left( 2a^{2}-\left( c^{2}+x^{2}+y^{2}\right) \right) ^{2}$

$4a^{4}-4a^{2}c^{2}-4a^{2}x^{2}-4a^{2}y^{2}+c^{4}+2c^{2}x^{2}+% \allowbreak 2c^{2}y^{2}+x^{4}+2x^{2}y^{2}+y^{4}-\left( c^{4}-2c^{2}x^{2}+2c^{2}y^{2}+x^{4}+2x^{2}y^{2}+y^{4}\right)=0$

$a^{2}-c^{2}-y^{2}+\left( \frac{b^{2}}{a^{2}}\right) x^{2}=0$

$\left( \frac{b^{2}}{a^{2}}\right) x^{2}-y^{2}=c^{2}-a^{2}=b^{2}$

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

The actual definition of hyperbola is that given the set of all points in the plane, the difference of whose distances(=2a , i've skipped this part of the proof because it is way too simple) from two fixed distinct is given postive constant that is less than the distance between the fixed points.

The above derivation is based on the definition.. I believe many people does not know the definition of it.. even for a simple parabola

DS' Random Work

Wednesday, November 24, 2010

Hyperbolas: Standard equation derivation

No comments:

Post a Comment