Let the locus of the mid points of the chords of circle x 2 + y − 1 2 = 1 drawn from the origin intersect the line x + y = 1 at P and Q . Then, the length of P Q is:

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Correct answer: A. 1 2

Let the mid point of the chords be x 1 , y 1 . The given equation of circle is x 2 + y - 1 2 = 1 . ⇒ x 2 + y 2 - 2 y = 0 ⇒ x x 1 + y y 1 - y + y 1 = x 1 2 + y 1 2 - 2 y 1 This locus passes through the origin, ⇒ 0 + 0 - 0 + y 1 = x 1 2 + y 1 2 - 2 y 1 ⇒ - y 1 = x 1 2 + y 1 2 - 2 y 1 ⇒ x 1 2 + y 1 2 - y 1 = 0 So, the locus is x 2 + y 2 - y = 0 It intersects with the line x + y = 1 . ⇒ 1 - y 2 + y 2 - y = 0 ⇒ 1 + y 2 - 2 y + y 2 - y = 0 ⇒ 2 y 2 - 2 y - y + 1 = 0 ⇒ 2 y y - 1 - y - 1 = 0 ⇒ 2 y - 1 y - 1 = 0 ⇒ y = 1 , 1

Let the locus of the mid points of the chords of circle x 2 + y − 1 2 = 1 drawn from the origin intersect the line x + y = 1 at P and Q . Then, the length of P Q is:

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