Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $4,$ and the slope of line segment $\overline{OQ}$ is $5.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$
Note: The point $(x,y)$ lies in the first quadrant if both $x$ and $y$ are positive.
Suppose P = (x, y). Since the slope of OP is 4, by slope formula, we have y−0x−0=4, which means y=4x.
So, P = (x, 4x) for some x. By the same argument, Q = (w, 5w) for some w.
Now, we use the condition OP=OQ to find a connection between w and x. Using distance formula to calculate OP and OQ, we have
√x2+(4x)2=√w2+(5w)2
17x2=26w2
x=√2617w
Now, the slope of PQ is 5w−4xw−x=5w−4⋅√2617ww−√2617w=5√17−4√26√17−√26
Rationalizing gives mPQ=(5√17−4√26)(√17+√26)17−26=−19+√442−9=19−√4429.
If you just need an approximate value, 19−√4429≈−0.2249.