The equations which have exactly one solution are:
Choice A: 
Choice C: 
Choice D: 
Explanation:
Choice A: 
Adding both sides by 52x, we have,

Subtracting both sides by 52,

Dividing both sides by -26,

Hence, the equation
has exactly one solution.
Thus, Choice A is the correct answer.
Choice B: 
Subtracting both sides by 52x,

Since, both sides are not equal.
It has no solution.
Thus, Option B is not the correct answer.
Choice C: 
Subtracting both sides by 78x,

Subtracting both sides by 52,

Dividing both sides by -20,

Hence, the equation
has exactly one solution.
Thus, Choice C is the correct answer.
Option D: 
Adding both sides by 78x,

Subtracting both sides by 52,

Dividing both sides by 136,

Hence, the equation
has exactly one solution.
Thus, Choice D is the correct answer.