Math question about circles and lines

Question

CIRCLES AND LINES PRE CALCULUS

I need help answering these two questions for a final exam. Please help

1.) Let A(1,3) and B(4,−2).  Find all points P on the y-axis such that AP⊥BP (that is ,∠APB= 90◦).

2.) Let A(2,4) and B(8,3).  Find the point P on thex-axis such that AP+PB is minimum.

Answer 1

1)

There are at least two approaches to this problem.

One approach is based on the fact that all the points P such that AP⊥BP lie

on the circle around AB as its diameter.

The other approach is based on the fact that all the points P such that AP⊥BP

satisfy the Pythagorean theorem, i.e., AP² + PB² = AB².

I do not know which one you are expected to use, but I will use the second.

Let P = (x,y).

Then the statement of the Pythagorean theorem is

(x - 1)² + (y - 3)² + (x - 4)² + (y + 2)² = (1 - 4)² + (3 + 2)²

You are supposed to find all such points P on the y-axis, that is when x = 0.

After substituting x = 0

(-1)² + (y - 3)² + (-4)² + (y + 2)² = (1 - 4)² + (3 + 2)²

After simplification you get the equation

y² - y - 2 = 0

It has two solutions y=2 and y=-1.

2)

All the points P on the x-axis are of the form (x,0).

For the quantity d = AP+PB to be minimized,

d² = (x - 2)² + (0 - 4)² + (x - 8)² + (0 - 3)²

= 2x² - 20x + 93

The value of x yielding minimum d also yields minimum d².

Therefore we can find it by setting the derivative of d² to 0.

4x - 20 = 0

x = 5