Question

Find the locus of the midpoint of all chords of the circle x^2+y^2-2x-2y=0 such that the pair of lines joining (0,0) and the point of intersection of the chords with the circles

Answer 1

Let a equation of line of slope m is passing through origin is y = mx.

line makes an angle $\theta$ with x axis. another line is also inclined to x axis with same angle but its slope will be -m.

now equation of other line will be y=-mx

let point of contact of line y=mx and y =-mx with circle be P and q respectively.

on solving y=mx and circle x² + y² - 2x - 2y = 0 we get,

P = [  2(1+m)/1+m², 2m(1+m)/1+m²   ]

and on solving y=-mx and circle x² + y² - 2x - 2y = 0 we get,

q = [  2(1-m)/1+m²,2m(m-1)/1+m²  ]

we have to find the locus of mid point of PQ because pq is chord of circle,

let X,Y is the mid point then, X=2/1+m²  and   Y=2m² /1+m²
on solving these we get, X+Y=2

hence, required answer is x + y = 2