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"Gererally" speaking cigar lighter sockets in cars are tip positive.
But to make sure you could hook up the current socket you have to a battery and hook a test light to the same battery's negative terminal and touch the end of the test light into the socket. If the test light lights up then the tip is positive.
You should be able to pick up a shorter socket from Repco as I got one for my Celestron CPC for around $15.
If you take the current one you have they should be able to match it up size-for-size.
I hope this helps.
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