Show that y=kx/(x-k) is symmetric with respect to y=x. Find the points of intersection of y with x-y= -2k...?

Question

...and find the slope of the tangent line at the point(s) of intersection.

Answer 1

Use Axis rotation equations to rotate your xy coordinates to pi/4 rad counter clock-wise.
Let new coordinates are x'y'.
Now you have x' axis along line y=x, and you have to show that graph is symmetric about x'-axis
You will see that the graph is hyperbola, with major axis along x'-axis.

Now show that replacing y' by -y' has no effect on equation

I'd not provide you solution in your benefit!!!

Answer 2

Swap the variables x and y and solve for y to get the inverse function.

x = ky / (y - k)
x(y - k) = ky
xy - kx = ky
xy - ky = kx
(x - k)y = kx
y = kx / (x - k)

Since the function is its own inverse it is symmetric to the line
y = x.

Answer 3

y=kx/(x-k)

x=ky/(y-k)
xy-xk=ky
xy-ky=xk
y(x-k)=xk
y=xk/(x-k)
So symmetric about y=x