Question

Find the locus of the middle points of the chords of the circle x² + y² = a² which pass through a given point (x₁ , y₁ ).

Answer 1

As line joining centre of given circle to the mid point of chord is perpendicular to the chord and hence product of their slope will be – 1. Therefore by considering mid point of chord as (α, β) and by finding their slope we will get required equation. 

Let M (α, β) be the middle point of any chord PQ through the given point (x₁ , y₁ ). The centre of the circle is O (0, 0). Clearly MO is perpendicular to PQ.

∴ the equation of the locus of M (α, β) is

x(x – x₁ ) + y(y – y₁ ) = 0 

Alternative 

The equation of chord when mid-point is known is 

T =S₁

Let the mid-point be (α β, )