If the distance from the point (5,k) to the line 8x - 15y + 8 + 0 is 20, find k. Recall that the formula to compute the distance between a point (x1,y1) and a line ax + by + c = 0 is

d= abs(ax1+by1+c) [over or divided by] square root(a^2+b^2)

To find the value of k, we need to use the formula for calculating the distance between a point and a line. The given formula is:

d = |ax1 + by1 + c| / √(a^2 + b^2)

In this case, the point is (5, k), and the line is represented by the equation 8x - 15y + 8 = 0. Comparing this equation to the standard form ax + by + c = 0, we have a = 8, b = -15, and c = 8.

Now we can substitute the values into the distance formula and solve for k:

d = |8(5) - 15(k) + 8| / √(8^2 + (-15)^2)

Given that the distance d is 20, we can rewrite the equation as:

20 = |40 - 15k + 8| / √(64 + 225)

To remove the absolute value, we can solve for both the positive and negative cases:

Positive case:
20 = (40 - 15k + 8) / √289
20√289 = 48 - 15k
20(17) = 48 - 15k
340 = 48 - 15k
-15k = 340 - 48
-15k = 292
k = 292 / -15

Negative case:
20 = -(40 - 15k + 8) / √289
20√289 = -(48 - 15k)
20(17) = -48 + 15k
340 = -48 + 15k
15k = 340 + 48
15k = 388
k = 388 / 15

Therefore, there are two possible values for k: k = 292 / -15 and k = 388 / 15.