Let the locus of the midpoints 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 $PQ$ is:
Step-by-Step Solution
Key Concept: If $(h,k)$ is midpoint of a chord drawn from origin $(0,0)$ on circle $x^2+(y-1)^2=1$: slope of chord $=k/h$. Chord is perpendicular to line joining centre $(0,1)$ to $(h,k)$: $(k/h)\cdot((k-1)/h)=-1\Rightarrow k(k-1)=-h^2\Rightarrow x^2+y^2-y=0$. Intersect with $x+y=1$.
Locus: $x^2+y^2-y=0$. With $x+y=1$: $P=(1/2,1/2)$ and $Q=(0,1)$. $PQ=\sqrt{1/4+1/4}=1/\sqrt{2}$.
Correct Answer: 1