if you have the points a(5,7) b(-1,-1) and p(x,y). how would you write an equaiton for all x and y for which pa is perpendicular to pb.

Best I can come up with is:

(py - by)/(px - bx) = -1 [(px - ax) / (py - ay)]

It relates the slopes of two perpendicular lines: each line's slope is the negative reciprocal of the others.

The problem arises when one of the slopes is 0, because then the (negative) reciprocal is some number over 0, which is undefined.

Best I can come up with is:

(py - by)/(px - bx) = -1 [(px - ax) / (py - ay)]

It relates the slopes of two perpendicular lines: each line's slope is the negative reciprocal of the others.

The problem arises when one of the slopes is 0, because then the (negative) reciprocal is some number over 0, which is undefined.

Anyway, pluggin in we get:

(py - (-1))/(px - (-1)) = -1 [(px - 5) / (py - 7)]

(py + 1)/(px + 1) = -1 [(px - 5) / (py - 7)]

To find the equation for all points P(x, y) for which PA is perpendicular to PB, we can use the concept of slopes. The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

slope = (y₂ - y₁) / (x₂ - x₁)

Let's calculate the slopes of PA and PB using the given points A(5, 7), B(-1, -1), and an arbitrary point P(x, y).

The slope of PA is:
slope_PA = (y - 7) / (x - 5)

The slope of PB is:
slope_PB = (y - (-1)) / (x - (-1)) = (y + 1) / (x + 1)

Since two lines are perpendicular if and only if the product of their slopes is -1, we can set the product of slopes_PA and slopes_PB equal to -1:

(slope_PA) * (slope_PB) = -1

[(y - 7) / (x - 5)] * [(y + 1) / (x + 1)] = -1

Now, by cross-multiplying and simplifying the equation, we can find the equation for all x and y for which PA is perpendicular to PB.