how do I activate airplay on my imac

Choose Apple Menu > System Preferences, then click Displays. If you don't see a “Show mirroring options in the menu bar when available” checkbox, your Mac doesn't support AirPlay.

Use AirPlay to display video from your Mac on an HDTV - Apple Support

Unfortunately a checkbox for “Show mirroring options in the menu bar when available” doesn't indicate that your Mac is able to support mirroring.

Which Mac do you have?

AirPlay Mirroring requires a second-generation Apple TV or later, OS X 10.8 or better and is supported on the following Mac models: iMac (Mid 2011 or newer), Mac mini (Mid 2011 or newer), MacBook Air (Mid 2011 or newer), and MacBook Pro (Early 2011 or newer). You can see which Mac you have and which operating system you are using by selecting 'About this Mac' from the Apple menu in the top left corner of your Mac and selecting the 'More Info' button.

AirPlay Mirroring from a Mac also requires a fairly robust network.

It is checked yet the AirPlay icon does not appear on the menu bar. When I uncheck it and then recheck it, there's a brief flash of an action on the menu bar which I cannot discern the details, and the icon does not appear.

Your Mac doesn't support mirroring, which requires a newer processor in order to do the video encoding in realtime for your full display to be sent to the Apple TV.

You could try a 3rd party program like airparrot but it's best to use the trial first to see how it works on your system.

If you have the option Preferences->Displays, make sure its checked, and the AirPlay icon should appear on the Menu bar at the top near the clock and other icons there.