Two points A and B move on the +ve direction of x-axis and y-axis respectively, such that OA + OB = a. Show that...

Prob 1. Two points A and B move on the +ve direction of x-axis and y-axis respectively, such that OA + OB = a. Show that the locus of the foot of the perpendicular from the origin O on the line AB is (x + y)(x² + y²) = axy.

Sol.

Let the equation of AB be

….(1)

given, a + b = α ….(2)

now, m_AB × m_OM = -1

⇒ ah = bk ….(3)

from (2) and (3),

∴ from (1)

as it passes through (h , k)

⇒ (h + k) (h² + k²) = αhk

∴ locus of (h , k) is (x + y) (x^2 + y^2) = αxy.