Multiple choice

The equation of the locus of the middle point of a chord of the circle ${ x }^{ 2 }+{ y }^{ 2 }=2(x+y)$ such that the point of lines joining the origin to the point of intersection of the chord and the circle are equally inclined to the x-axis is

  1. $x + y = 2$
  2. $x - y = 2$
  3. $2x - y = 1$

  4. None of these
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A Correct answer
Explanation

The locus of the midpoint of a chord is found by using the condition that the chord is bisected at the point (h, k). The condition of equal inclination to the x-axis for the lines joining the origin to the intersection points leads to the equation x + y = 2.

AI explanation

The given circle equation x^2 + y^2 = 2(x + y) represents a circle passing through the origin with its centre at (1, 1) and radius sqrt(2). Let the middle point of the chord be (h, k), so the equation of the chord is T = S1, which expands to xh + yk - (x + h) - (y + k) + h^2 + k^2 - 2h - 2k = 0. Since the lines joining the origin to the intersection points are equally inclined to the x-axis, the combined equation of these lines lacks the xy term. Substituting y = mx into the homogeneous part of the chord equation and enforcing the condition that the product of the slopes is -1 forces the sum of the roots to zero, yielding the locus x + y = 2.

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