Date: 202009
Local time of day: 1801-2400
Location: ZZZ.TRACON, US
Light: Daylight
Aircraft: Dassault-Breguet Undifferentiated or Other Model
Operator: Corporate
Operating under: Part 135
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class E ZZZ
ATC / advisory: TRACON ZZZ
Second aircraft: Small Aircraft
Reporter organization: Government
Reporter function: Approach
Reporter qualification: Air Traffic Control Fully Certified
Human factors: Communication Breakdown; Distraction; Situational Awareness; Confusion
Anomaly: ATC Issue All Types; Conflict NMAC; Conflict Airborne Conflict; Deviation / Discrepancy - Procedural FAR; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Automation Aircraft RA
When detected: In-flight
Result: Air Traffic Control Issued Advisory / Alert; Air Traffic Control Issued New Clearance; Flight Crew Took Evasive Action; Flight Crew FLC complied w / Automation / Advisory
Contributing factors: Company Policy; Human Factors
Primary problem: Human Factors

Narrative: I was working a satellite sector and had an IFR release off of ZZZ airport for [Aircraft X]. [Aircraft X] was filed for a west bound SID climbing into the flight levels. When [Aircraft X] departed I got an automated point out with Departure Control which was combined at Arrival Sector. The automated point out was approved. I climbed [Aircraft X] to 16000 ft. and sent the aircraft on course. A few seconds later; Arrival sector called me on the landline to combine Departure Control and Arrival to me. During the brief I was told about a VFR PA31 (Aircraft Y) conducting a photo mission at 11500 ft. I was briefed that they were working southwest to the northeast. I failed to see that [Aircraft X] and [Aircraft Y] were on converging courses. [Aircraft X] was climbing; at what I thought; was an expeditious pace. The Arrival controller called me on the landline while doing his sector briefing overlap and asked if I was watching [Aircraft X] and [Aircraft Y]; which I was not. At that point; the Mode C of [Aircraft X] was 11000 ft. and the two aircraft were right at 3 miles apart. I told [Aircraft X] to expedite their climb for traffic; and at that point; [Aircraft X] stated that they were descending for a TCAS RA. [Aircraft X] said that they have the aircraft in sight and are descending. I did not issue a traffic call or traffic alert since [Aircraft X] stated they had the aircraft in sight. I issued traffic to the photo mission aircraft; [Aircraft Y] twice; but [Aircraft X] responded both times thinking I was talking to them.My supervisor stated that the two aircraft got 2.1 miles and 500 ft. apart. I yelled out that I had an aircraft respond to a TCAS RA; but the Supervisor did not hear me; and I did not make sure that the management knew. The pilot then climbed back to 16000 ft. and I transferred communications to Center. When the aircraft was in their destination TRACON's airspace; I was told that the pilot told the approach controller that he almost had a near midair. The pilot called our TRACON and did not mention a near midair; and my supervisor explained that I made a mistake and missed the traffic.Better situational awareness on my end. I was not sure how far the photo mission flew before they turned around; so I need to clarify that during the briefing.




Synopsis: TRACON Controller reported NMAC with two aircraft within their sector.