knight247 wrote:Hey Brent,
Appreciate ur response. Is there any way that you could help prove that the slope of line y=mx+b reflected over y=x is 1/m? I'm having a hard time understanding how that is so. Or maybe if you could show me any links where I could learn that. Also, Are there any other similar properties or corollaries related to reflections over y=x that are important to know. I mean, besides the two mentioned here. Thanks
Aside: This is pretty esoteric stuff. You certainly don't need to know this rule or this proof for the GMAT.
That said, here's a pretty rudimentary proof:
If the line y=mx+b is reflected across the line y=x, the equation of the reflected line is x=my+b (to find the equation of any reflected line across y=x, just switch the x's and y's)
Now let's find 2 points on the reflected line x=my+b
When y=0, we get x=m(0)+b, which equals b.
So, one point on the line is (b,0)
When y=1, we get x=m(1)+b, which equals m+b.
So, a second point on the line is (m+b,1)
When we use the slope formula to find the slope between these two points, we get:
Slope = [1-0]/[(m+b)-(b)] = 1/m
Cheers,
Brent
