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Precalculus Mathematics Homework Help
Hyperbola Focus Length Greater than Semi-Major Axis: Is it a Necessity?
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[QUOTE="bananabandana, post: 4955479, member: 490819"] [h2]Homework Statement [/h2] Why is it necessarily true that for a hyperbola, the focus length, ##f ## has got to be greater than the semi-major axis , ## a## - ## f >a ## ? [h2]Homework Equations[/h2] - [h2]The Attempt at a Solution[/h2] I needed to derive the cartesian equation of a hyperbola with centre at ## (\alpha,\beta) ## and foci along the ##\alpha - ## axis. using the definition that the difference between the distances to the two foci is a constant. After some algebra, I just got back the equation of an ellipse. $$ \frac{(x-\alpha)^{2}}{a^{2}}+\frac{(y-\beta)^{2}}{a^{2}-f^{2}}=1 $$ Of course I realize that this is also the equation of a hyperbola, under the condition that ## f>a ## so that : $$ \frac{(x-\alpha)^{2}}{a^{2}}-\frac{(y-\beta)^{2}}{a^{2}-f^{2}}=1 $$ But I'm stuck at proving that this condition is, in general true, just from the definition given. Could someone give me a hint? Thanks! [/QUOTE]
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Precalculus Mathematics Homework Help
Hyperbola Focus Length Greater than Semi-Major Axis: Is it a Necessity?
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