What is the equation of the locus of the mid-point of the line segment obtained by cutting the line x + y = p, (where p is a real number) by the coordinate axes ?

Concept:

The distance between two points in a two-dimensional plane is given by \(\displaystyle d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Calculation:

Given:

The equation of the line is x + y = p. 

Let the line cut the coordinate axes at A and B and if C(h, k) be the midpoints of AB, then

Now the intercept made by the line on the coordinate axes is p.

Point C(h, k) is equidistant from the points A(p, 0) and B(0, p)

As C is the midpoint, AC = BC

⇒ \(\displaystyle \sqrt{(h-p)^2+(k-0)^2}=\sqrt{(h-0)^2+(k-p)^2}\)

Squaring both sides, we get,

⇒ (h - p)2 + k2 = h2 + (k - p)2

⇒ h² + p2 - 2hp + k2 = h2 + k2 + p2 - 2pk

⇒ - 2hp = - 2pk

⇒ h = k

Now, to obtain the locus of the mid-point replace (h, k) with (x, y),

⇒ x = y

∴ x - y = 0 is correct.