+1234 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.
+129847 Construct a circle centered at the origin with a radius of 1
The equation of this circle is
x^2 + y^2 = 1 (1)
Let P, Q lie on this circle
Line through OP has the equation
y = 4x
Sub this into (1) to find the x coordinate of P
x^2 + (4x)^2 =1
17x^2 = 1
x^2 = 1/17
x =1/sqrt 17
y = 4/sqrt 17
P = (1/sqrt 17, 4/sqrt 17)
Similarly, line through OQ has the equation
y =5x
Sub this into (1) to find the x coordinate of Q
x^2 + (5x)^2 =1
26x^2 = 1
x^2 = 1/26
x =1/sqrt 26
y = 5/sqrt 26
Q = ( 1/sqrt 26, 5/sqrt 26)
Slope of PQ =
[ 4/sqrt 17 - 5/sqrt 26 ] / [ 1/sqrt 17 - 1/sqrt 26 ] = [ 5 sqrt 26 - 20 ] / [sqrt 26 - 4 ] =
[ 4sqrt 26 - 5sqrt 17 ] / [ sqrt 26 - sqrt 17 ] =
[ 4sqrt 26 - 5sqrt 17 ] [ sqrt 26 + sqrt 17 ] / 9 =
[4 * 26 - sqrt (442) - 85 ] / 9 =
[19 - sqrt 442 ] / 9 ≈ -0.225
