LESS THAN STANDARD SEPARATION BETWEEN GA-TWIN AND CORP ACFT. OPERATIONAL ERROR.

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I WAS WORKING THE S SECTOR OF OUR APCH AIRSPACE. NOT WORKING A LOT OF AIRPLANES; BUT ENOUGH TO KEEP ME BUSY. THE N CTLR ASKED THE COORDINATOR IF HE COULD ENTER MY AIRSPACE WITH ACFT X AT 5000' ENRTE TO A SATELLITE ARPT. NORMALLY I DON'T APPROVE THAT OPERATION BECAUSE IT GOES OVER THE TOP OF THE MAIN ARPT AT THE SAME ALT AS OUR IFR DOWNWINDS. BUT; N WAS TRAPPED ON TOP OF A VFR AT 4500'; SO I APPROVED IT. WHEN ACFT X CHKED IN AT 5000' ON ONE FREQ; A SATELLITE DEP CHKED IN ON ANOTHER ONE. INSTEAD OF ATTEMPTING TO DSND THE 5000' ACFT TO A LOWER ALT; I TALKED TO THE SATELLITE DEP INSTEAD TO GET HIM CLBING. WHILE CONCENTRATING ON THE SATELLITE DEP AND OTHER SATELLITE ARRS; I FAILED TO SEE THE 5000' ACFT COME OUT OF THE MAIN BANG OF THE RADAR ANTENNA; NOT TO MENTION HE WAS OVERLAPPED ON HIS DATA BLOCK WITH ANOTHER ACFT. I HAD CORP Y DOWNWIND AT 5000' TALKING TO THE FINAL CTLR AT THE SAME TIME. THE FIRST TIME I REALIZED THE PROB WAS WHEN THE C/A WENT OFF. TOO LATE TO SAVE THE CONFLICT; BUT IN ENOUGH TIME TO TURN MY ACFT X AWAY FROM CORP Y; THUS AVOIDING A NMAC. CONTRIBUTING FACTORS WOULD BE MY FAILURE TO KEEP UP A PROPER SCAN; WRONG PRIORITY OF DUTIES; WORKING COMBINED SECTORS AND POSSIBLY THE FAILURE OF THE COORDINATOR TO ALSO REALIZE THE SITUATION. SITUATION ALSO OCCURRED BECAUSE I APPROVED A SITUATION THAT I WOULD NOT NORMALLY DO. ONE WAY TO PREVENT THIS OCCURRENCE FROM HAPPENING AGAIN WOULD BE TO BE MORE AWARE OF POTENTIAL CONFLICTS; DON'T GET OUT OF A ROUTINE IF AT ALL POSSIBLE. IT WOULD ALSO BE NICE IF THE ARTS IIIA C/A WOULD WORK BETTER THAN IT DOES. RIGHT NOW ABOUT THE ONLY TIME IT GOES OFF IS WHEN THE CONFLICT HAS ALREADY OCCURRED. IT WOULD BE GREAT IF IT WOULD PREDICT THE CONFLICT IN ENOUGH TIME TO BE ABLE TO RESOLVE THE SITUATION.