A line \((\sin\theta)x + (\cos\theta)y = \sin 2\theta\) cuts the coordinate axes at points \(P\) and \(Q\). Let \(M\) be the midpoint of the line segment \(PQ\). What is the distance of \(M\) from the origin?
\(1\)
Putting \(y=0\) gives the \(x\)-intercept \(P=(2\cos\theta,0)\), and putting \(x=0\) gives the \(y\)-intercept \(Q=(0,2\sin\theta)\), using \(\sin2\theta=2\sin\theta\cos\theta\). The midpoint is \(M=(\cos\theta,\sin\theta)\), so its distance from the origin is \(\sqrt{\cos^2\theta+\sin^2\theta}=1\).
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