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GeometryQuestion and Answers: Page 91

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If p = cos θ + i sinθ and q = cos φ + i sin φ Show that: (i) ((p − q)/(p + q)) = i tan (((θ − φ)/2)) (ii) (((p + q)(pq − 1))/((p − q)(pq + 1))) = ((sin θ + sin φ)/(sin θ − sin φ))

$$\mathrm{If}\:\:\:\mathrm{p}\:\:=\:\:\mathrm{cos}\:\theta\:+\:\mathrm{i}\:\mathrm{sin}\theta\:\:\:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\mathrm{q}\:\:=\:\:\mathrm{cos}\:\phi\:+\:\mathrm{i}\:\mathrm{sin}\:\phi \\ $$$$\mathrm{Show}\:\mathrm{that}: \\ $$$$\left(\mathrm{i}\right)\:\:\:\:\:\:\frac{\mathrm{p}\:−\:\mathrm{q}}{\mathrm{p}\:+\:\mathrm{q}}\:\:=\:\:\mathrm{i}\:\mathrm{tan}\:\left(\frac{\theta\:−\:\phi}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{ii}\right)\:\:\:\frac{\left(\mathrm{p}\:+\:\mathrm{q}\right)\left(\mathrm{pq}\:−\:\mathrm{1}\right)}{\left(\mathrm{p}\:−\:\mathrm{q}\right)\left(\mathrm{pq}\:+\:\mathrm{1}\right)}\:\:=\:\:\frac{\mathrm{sin}\:\theta\:+\:\mathrm{sin}\:\phi}{\mathrm{sin}\:\theta\:−\:\mathrm{sin}\:\phi} \\ $$

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A and B are two points from the plan (P) with AB=4 define and draw the locus of points M ∈(P) wich verify MA +MB =8 .

$${A}\:{and}\:{B}\:{are}\:{two}\:{points}\:{from}\:{the}\:{plan}\:\left({P}\right)\:{with}\:{AB}=\mathrm{4}\:{define}\:{and}\:{draw} \\ $$$${the}\:{locus}\:{of}\:{points}\:{M}\:\in\left({P}\right)\:{wich}\:{verify}\:\:\:{MA}\:+{MB}\:=\mathrm{8}\:. \\ $$

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