A curve is given the equation xy=25 and the equation of a line y=mx+b. i. Express m in terms of b given that the line and the curve have one and only one point of intersection.
ii. If the line intersects the curve in two distinct points, find the set of values of b, given m=-5

Below in bold.

Step-by-step explanation:

i, This is a rectangular hyperbola with symmetry about the line y = -x

   If it touches the graph at one point only then

y = 25/x

y = mx + b so

25/x = mx + b

mx^2 + bx + 25 = 0

Using the quadratic formula:

x = (-b + sqrt(b^2 - 25*4*m)) / 2m

Now there's only one value of x so

b^2 - 25*4*m = 0

b^2 = 100m

m = b^2/100.

m = +/-b/10.

ii  In this case b^2 > 100m

If m = |-5|

b^2 > 500

b > +/-√500

So b > √500 or b < -√500.