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The first curve is a circle of radius $1$ centred at $(2,1)$. A line through the origin with equation $kx-y=0$ intersects it precisely when the perpendicular distance from that centre is at most the radius. The distance from a point to a line givesHence the full parameter range isThe two endpoint lines are tangent, at $(2,0)$ and $(6/5,8/5)$ respectively; interior values give two real intersections. Both endpoints are therefore included.

The original PDF contains the product $\sin x(1+4\cos x)$; the transcription drops the $x$ after the first sine. Using $2\sin x\cos x=\sin 2x$ gives the finite Fourier sine seriesThus $b_1=1$, $b_2=2$, and all other sine coefficients vanish. No cosine or constant terms are present.