If (k,0), (0d) and (1,1) are collinear then find 1/k + 1/d = ---------

To determine if the points (k,0), (0,d), and (1,1) are collinear, we need to check if the slope between any two pairs of points is the same.

The slope between two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)

Let's calculate the slopes of the line segments:

1) Slope between (k,0) and (0,d):
slope1 = (d - 0) / (0 - k) = -d/k

2) Slope between (0,d) and (1,1):
slope2 = (1 - d) / (1 - 0) = 1 - d

If the points are collinear, then the slopes between any two pairs of points should be equal. Therefore, we can set slope1 equal to slope2 and solve for d and k:

-d/k = 1 - d

To simplify the equation, we can multiply both sides by k:
-d = k - dk

Next, we can isolate d:
d - dk = -d
d(1 - k) = -d
d = -d / (1 - k)

Now, we can substitute the value of d in the expression 1/k + 1/d:

1/k + 1/(-d / (1 - k))
1/k - (1 - k)/d
[(1 - (1 - k))/k] - (1 - k)/(-d)
[(1 - 1 + k)/k] - (1 - k)/(-d)
(k/k) - (1 - k)/(-d)
1 - (1 - k)/(-d)
1 + (1 - k)/d
1 + 1/d - k/d

Therefore, 1/k + 1/d = 1 + 1/d - k/d.