Both of the points P(17, −20) and Q(25, t) are in the xy-plane. Which

No it's not correct. Thank you noticing! It's t > -18. Edited. Thank you!

actually Bunuel with t > -18t > -18 can also be checked that's valid only for 0<t< 1

We need unique value of t.
Statement I: Since the slope is +ve, we can find the value of 't'
Statement III : Can find the value of t using the midpoint formula

However, statement II will not give us a unique value of t since a quadratic equation is being formed.
IMO D should be the answer.

I almost took 2 mins to solve this question, need to improve.

I. The slope of the line that goes through the points P(17, − 20) and Q(25, t) is 3/4
slope= (t+20)/(25-17)=3/4 or, t= -14>-18. Correct.

II. The distance between the points P(17, − 20) and Q(25, t) is 10.
Solving equation, t+20=+-6, t=-26<18 and t=-14>-18.Incorrect.

III. The point Q(25, t) is the midpoint of the line segment whose endpoints are P(17, −20) and R(33, 3t + 34).
solving for y -coordinate only, 3t+14= 2t or, t= -14>-18 Correct.

So, I think D.

Any quant expert if you can help, IanStewart, MartyTargetTestPrep MathRevolution

T can be positive or negative hence I solved via taking t and -t which gave me
t = -14 or t= 26 for first statement using slope formula
t = -14 or t = 14 for 3rd statement using midpoint formula
I got the answer right, but is my approach wrong ? As other 3 co-ordinates are fix, there can be just one value of t so should I just use t instead of 2 cases i.e t and -t ? Or we are not considering -t because it will be positive as t gives negative value, which has to be greater than -18 (q stem's value)

Yes, your approach is not correct, and this is a very important point, because you'll get a lot of algebra questions wrong if you solve them this way. I'll use a simpler example -- say we have this equation:

x + 2 = -3

Here x can be negative (it must be negative, in fact), but we do not then solve this equation by replacing "x" with "-x". We just solve for x:

x = -5

and at the end, we learn that x stands for a negative value. The same is true in this question; you should not replace "t" with "-t" just because t can be negative. If you do that, you're solving for the negative of t, not for t itself (and when you did that using the slope formula, you should have gotten 14, the negative of the right answer, not 26, so you made an arithmetic mistake somewhere).

It's possible you were thinking about absolute value equations when you solved this way. When we see an equation like |x| = 5, one way to solve is to replace "|x|" with x, and with -x, and solve the two equations that result, but we do that because absolute value changes the sign of a negative number. That's not something we do just because x can be negative.