2 points A and B are moving on the x-axis and y-axis separately. AB=20 units. Find the equation of the locus P where P is the midpoint of AB.
If A and B are moving on the x-axis and y-axis in such a way that AB = 20 Units ( Constant) then the middle point of AB lies on a circle with centre at the origin and radius r = 10 units= AB/2 ...This can be easily found from the right angle triangle OAB with OM = r (M is the mid point of AB ) to be equal to AB/2 =10 units...
However ,if you take another point other than the middle point of AB then the locus of this point is an ellipse.
This is the well known theorem of Frans Van Shooten in Geometry which is also valid in the case that the two lines ( x-axis and y-axis ) are not perpendiculars.
The french Mathematician Victor Amedee Mannheim calculated the lengths of the two axes of that ellipse in a work published in 1850 ( Nouvelles Annales des Mathematiques ,page 419 ,1850) .
Let the point on y axis A be ( 0, y) and point B on x axis be ( x,0). Let the middle point be ( h,k )
Now by distance formula AB^2 = x^2+y^2 = 20^2 ----- (1)
that is x^2 + y^2 = 400.
Since (h,k) is mid point of (x,0) and (0,y) thus h = x/2 and k=y/2
That gives x=2h and y = 2k. Plugging these values in (1) we get h^2 + k^2 = 100.
For equation of locus replace h by x and y by k we get
x^2 + y^2 = 100
That is a circle with radius 10 and center at ( 0,0)