OA & OB are perpendiculars done in such a manner that OA+OB=8 Find equation of locus of middle point of AB?

Answer 1

Let OA be along the X axis. Then OB will be along the Y axis.

Let P (x,y) be the midpoint of AB. Then A will bw (2x,0) and B wil be (0,2y).

Now, 2x + 2y = 8.
Hence x + y = 4, or y = -x +4.

This is the equation of the locus of the midpoint of AB. It is a straight line with slope -1 and passing through (0,4).

Answer 2

Take OA and OB as x and y axis respectively A(a,0) B(0,b)

a+b=8 Midpoint (x=a/2 ,y=b/2) but b/2= 4-a/2 So the equation is

y=4-x a straight line with slope -1and y intercept 4

Answer 3

In essence you are breaking a line segement of length 8 into two segments, one horizontal and one vertical. If the length of the horizontal segement is x, then the vertical segment is 8 - x. If we call the vertical segment y then we have:

y = 8 - x

This is a straight line with slope -1. If you restrict x and y to positive values, it is a line segment of length 8√2.