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GeometryQuestion and Answers: Page 94

Question Number 51905    Answers: 2   Comments: 0

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))

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

Question Number 51884    Answers: 1   Comments: 4

Question Number 51849    Answers: 3   Comments: 3

Question Number 51700    Answers: 1   Comments: 1

Question Number 51658    Answers: 3   Comments: 7

Question Number 51636    Answers: 1   Comments: 4

Question Number 51644    Answers: 0   Comments: 0

Question Number 51605    Answers: 0   Comments: 0

Question Number 51558    Answers: 1   Comments: 3

Question Number 51549    Answers: 0   Comments: 0

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 .

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

Question Number 51502    Answers: 0   Comments: 1

Question Number 51466    Answers: 1   Comments: 5

Question Number 51245    Answers: 2   Comments: 0

Question Number 51141    Answers: 1   Comments: 0

Question Number 51088    Answers: 2   Comments: 1

Question Number 50829    Answers: 2   Comments: 1

Question Number 50762    Answers: 1   Comments: 1

Question Number 50674    Answers: 2   Comments: 3

Find the maximum area of a triangle inscribed in an ellipse with parameters a and b.

$${Find}\:{the}\:{maximum}\:{area}\:{of}\:{a}\:{triangle} \\ $$$${inscribed}\:{in}\:{an}\:{ellipse}\:{with}\:{parameters} \\ $$$${a}\:{and}\:{b}. \\ $$

Question Number 50577    Answers: 1   Comments: 1

Question Number 50561    Answers: 3   Comments: 1

Question Number 50460    Answers: 2   Comments: 2

Question Number 50352    Answers: 1   Comments: 5

The distances from a point to the sides of a triangle are p,q,r. Find the maximum (or minimum) area of the triangle, if it exists. Assume r≤q≤p.

$${The}\:{distances}\:{from}\:{a}\:{point}\:{to}\:{the}\:{sides} \\ $$$${of}\:{a}\:{triangle}\:{are}\:{p},{q},{r}.\:{Find}\:{the}\: \\ $$$${maximum}\:\left({or}\:{minimum}\right)\:{area}\:{of}\:{the} \\ $$$${triangle},\:{if}\:{it}\:{exists}. \\ $$$${Assume}\:{r}\leqslant{q}\leqslant{p}. \\ $$

Question Number 49987    Answers: 1   Comments: 4

Question Number 49984    Answers: 1   Comments: 1

Question Number 49970    Answers: 2   Comments: 1

Question Number 49830    Answers: 3   Comments: 1

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