i have an old radio shack dx-100 how can i can hooked this to it ?

From looking at some old manuals online, the dx-100 has a screw terminal for an external antenna, not socket for a plug like the reel antenna in question. I think you will have success if you cut off the plug, strip the end of the wire from insulation, and use the screw terminal. You will not need any special tools - just a good sharp knife.

Can this antenna work and improve am/fm radio reception?

It will greatly improve FM reception. AM uses a different type of antenna but a slight improvement may be noticed on AM.

I already have the reel antenna. How do I order just the adapter clip alone?

I don't think you can, there is no listing for parts, just solder a small clamp and that's it.

Is there a way (a cable?) to extend the distance from the antenna to the radio?

Yes you can use an audio cable extension https://www.amazon.com/gp/product/B075SV6XZY/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1

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XHDATA AN-80 Shortwave Reel Antenna FM SW External Antenna

Product ID: 284479675

Details

Antenna
Radio
Brand
XHDATA
Color
Black
Impedance
75 ohm
Manufacturer
XHDATA
Maximum Range
23 feet

🎒ultra-portable & lightweight

📡720 cm max length

🔧spring-loaded clip for easy setup

S$24

Duties & taxes included

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Description

📡 Catch Every Wave, Wherever You Go!

TRAVEL READY DESIGN - Compact and lightweight, fits right in your pocket—perfect for hikes, fishing trips, and urban escapes.
PLUG PLAY SIMPLICITY - Effortless setup with a spring-loaded clip and easy mounting—get crystal-clear signals in minutes.
UNIVERSAL COMPATIBILITY - Works seamlessly with any FM/SW radio, even those without antenna jacks, thanks to its smart clip design.
EXTEND YOUR SIGNAL REACH - A whopping 23 feet of antenna length boosts your FM and shortwave reception beyond limits.
RELIABLE SUPPORT WARRANTY - Backed by a one-year warranty and responsive customer service, ensuring your signal stays strong.

The XHDATA AN-80 is a 23-foot long, portable shortwave and FM external antenna designed to significantly enhance radio reception. Its lightweight, pocket-sized design and spring-loaded clip make it compatible with virtually all FM/SW radios, even those lacking antenna jacks. Ideal for outdoor enthusiasts and signal seekers, it offers easy setup and reliable performance backed by a one-year warranty.

Specifications

ASIN	B08VD6T4YK
AntennaDescription	Radio
Best Sellers Rank	#13 in Radio Antennas
Brand	XHDATA
Color	Black
Customer Reviews	4.3 4.3 out of 5 stars (1,439)
Impedance	75 ohm
Item Dimensions L x W x H	3.14"L x 1.18"W x 3.7"H
Item Highlight	Improve FM/Shortwave Signal Reception, suitable for FM Shortwave Radio
Item Type Name	FM Antenna SW Antenna
Manufacturer	XHDATA
Maximum Range	23 feet
Mfr Part Number	AN80
Model Number	AN-80
Number of Channels	2
Number of Items	1
Number of Packs	1
UPC	767065103953
Unit Count	1 Count
Warranty Description	The Antenna has a one-year warranty, for the details, please contact customer service. If you have any questions, contact us without hesitation. We'll try our best to solve the problem for you.

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