Apple TV and MacMini

1) I gather the AppleTV can be connected to a MacMini via a microusb lead, is that correct ? What does this lead look like ?

2) Also, I want to connect the Macmini to a monitor, but gather the only way to do this is via a HDMI connection, is that correct ? As the Macmini does not have a VGA connector.
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2). No.

Depending on the age of your Mac Mini, it will have either a mini display port or one or more Thunderbolt ports. (These port are physically identical).

You can connect your mini to a monitor using either a mini display port to VGA connector or a mini display port to DVI connector:

As well, some older minis had a mini-dvi port.
1) Depends on why you want to connect it. You only need to connect by USB to service the AppleTV, say for instance to restore it. Normally you would connect by ethernet or wifi.

If you do need to do something that requires the mini-usb cable, you can find one here:

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