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Homework Statement
Why is it necessarily true that for a hyperbola, the focus length, ##f ## has got to be greater than the semimajor axis , ## a##  ## f >a ## ?
Homework Equations

The Attempt at a Solution
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 realise 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!