Answer:
8.75
Step-by-step explanation:
You want the greatest possible x-coordinate of point A such that for any point B on the y-axis between (-24) and 25, the segment AB has a common point with the line segment PQ, where P(5, -12), Q(5, 9).
Geometry
If we designate the points on the y-axis as X(0, -24) and Y(0, 25), we find that point A must lie in the triangle bounded by lines XY, XP, and YQ. The x-coordinate of the point of intersection of lines XP and YQ will be the greatest x-coordinate that point A can have.
Equations
The line through points (x1, y1) and (x2, y2) can be written as ...
(y2 -y1)(x -x1) -(x2 -x1)(y -y1) = 0
This equation is readily simplified to general form ax+by+c = 0.
Line XP
(-12-(-24))(x -0) -(5-0)(y -(-24)) = 0
12x -5y -120 = 0
Line YQ
(9 -25)(x -0) -(5 -0)(y -(25)) = 0
-16x -5y +125 = 0
We want the value of x that satisfies these equations. That can be found by subtracting the second equation from the first:
(12 -5y -120) -(-16x -5y +125) = (0) -(0)
28x -245 = 0
x = 245/28 = 8.75
The greatest x-coordinate is x = 8.75.
