Originally Posted by

**free_to_fly** *pokes title*

The question is:

P is any point on the hyperbola with the equation $\displaystyle \

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

\$. S is the focus $\displaystyle \

(ae,0)

\$ and S' is the focus $\displaystyle \

( - ae,0)

\$, where e is the eccentricity. Show that $\displaystyle \

\left| {SP - S'P} \right| = 2a \ $ Mr F says: This is wrong. It should be 2ae. You can see this for yourself by taking P(a, 0) and P'(-a, 0) ...... It's $\displaystyle \

\left| {SP + S'P} \right| = 2a \ $ ....

My working:

I've found the length ofSP and S'P using good old Pythagoras

$\displaystyle \

SP = \left[ {(ae - x)^2 + (\frac{{ - b}}{a}\sqrt {x^2 - a^2 } )^2 } \right]^{\frac{1}{2}}

\$, so

$\displaystyle \

SP = a - x\sqrt {\frac{{b^2 }}{{a^2 }} + 1}

\$

and similarly $\displaystyle \

S'P = a + x\sqrt {\frac{{b^2 }}{{a^2 }} + 1}

\$, which gives me

$\displaystyle \

\left| {SP - S'P} \right| = 2x\sqrt {\frac{{b^2 }}{{a^2 }} + 1}

\$

If one of the signs changes in SP or S'P then I'd get the right answer. Can someone please check if I made a mistake with the signs? Thanks.