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