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Let Γ be a hyperbola with foci F_1 and F_2 , eccentricity e. M is an arbitrary point on Γ. Let x=∠MF_1 F_2 , y=∠MF_2 F_1 Prove that ((∣ cos x − cos y ∣)/(1 − cos x cos y)) = ((2e)/(e^2 +1)).

$$\mathrm{Let}\:\Gamma\:\mathrm{be}\:\mathrm{a}\:\mathrm{hyperbola}\:\mathrm{with}\:\mathrm{foci}\:{F}_{\mathrm{1}} \:\mathrm{and}\:{F}_{\mathrm{2}} ,\: \\ $$$$\mathrm{eccentricity}\:{e}.\:{M}\:\mathrm{is}\:\mathrm{an}\:\mathrm{arbitrary}\:\mathrm{point}\:\mathrm{on}\:\Gamma. \\ $$$$\mathrm{Let}\:{x}=\angle{MF}_{\mathrm{1}} {F}_{\mathrm{2}} ,\:{y}=\angle{MF}_{\mathrm{2}} {F}_{\mathrm{1}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\frac{\mid\:\mathrm{cos}\:{x}\:−\:\mathrm{cos}\:{y}\:\mid}{\mathrm{1}\:−\:\mathrm{cos}\:{x}\:\mathrm{cos}\:{y}}\:=\:\frac{\mathrm{2}{e}}{{e}^{\mathrm{2}} +\mathrm{1}}. \\ $$

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If α, β, γ, δ are the roots of x^4 + x^3 + x^2 + x + 1 = 0 then find α^(2021) + β^(2021) + γ^(2021) + δ^(2021) .

$$\mathrm{If}\:\alpha,\:\beta,\:\gamma,\:\delta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${x}^{\mathrm{4}} \:+\:{x}^{\mathrm{3}} \:+\:{x}^{\mathrm{2}} \:+\:{x}\:+\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{find} \\ $$$$\alpha^{\mathrm{2021}} \:+\:\beta^{\mathrm{2021}} \:+\:\gamma^{\mathrm{2021}} \:+\:\delta^{\mathrm{2021}} \:. \\ $$

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Question Number 215775 Answers: 1 Comments: 1

∫_0 ^( s) (√(1−s^2 ))(1+(√(1−((x/s))^2 ))−(√(1−x^2 )))dx

$$\:\:\underset{\mathrm{0}} {\overset{\:\:{s}} {\int}}\sqrt{\mathrm{1}−{s}^{\mathrm{2}} }\left(\mathrm{1}+\sqrt{\mathrm{1}−\left(\frac{{x}}{{s}}\right)^{\mathrm{2}} }−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 215774 Answers: 1 Comments: 0

let's say there are two circles with their centers A1 and A2 and their radii are equal to r1 and r2 ,let n be the distance between the centers of the two circles and n

$$ \\ $$let's say there are two circles with their centers A1 and A2 and their radii are equal to r1 and r2 ,let n be the distance between the centers of the two circles and n<r1+r2. Is there a way to find how much is the area that is formed when the two circles are overlapping ?

Question Number 215769 Answers: 1 Comments: 1

In △ABC, it is given that AC⊥CB, CD⊥AB, and CD = 12, AC = BC + 5. Please solve for the value of BC using a purely geometric method.

Question Number 215760 Answers: 1 Comments: 0

Let g(x) = 3f((x/3)) + f(3 − x) and f ′′(x) > 0 for all x ∈ (0, 3). If g is decreasing in (0, α) and increasing in (α, 3) then find 8α.

$$\mathrm{Let}\:{g}\left({x}\right)\:=\:\mathrm{3}{f}\left(\frac{{x}}{\mathrm{3}}\right)\:+\:{f}\left(\mathrm{3}\:−\:{x}\right)\:\mathrm{and}\: \\ $$$${f}\:''\left({x}\right)\:>\:\mathrm{0}\:\mathrm{for}\:\mathrm{all}\:{x}\:\in\:\left(\mathrm{0},\:\mathrm{3}\right).\:\mathrm{If}\:{g}\:\mathrm{is}\: \\ $$$$\mathrm{decreasing}\:\mathrm{in}\:\left(\mathrm{0},\:\alpha\right)\:\mathrm{and}\:\mathrm{increasing}\:\mathrm{in} \\ $$$$\left(\alpha,\:\mathrm{3}\right)\:\mathrm{then}\:\mathrm{find}\:\mathrm{8}\alpha. \\ $$

Question Number 215748 Answers: 1 Comments: 0

lim_(n→∞) [Π_(k=1) ^(n+1) Γ((1/k))]^(1/(n+1)) −[Π_(k=1) ^n Γ((1/k))]^(1/n)

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left[\underset{{k}=\mathrm{1}} {\overset{{n}+\mathrm{1}} {\prod}}\Gamma\left(\frac{\mathrm{1}}{{k}}\right)\right]^{\frac{\mathrm{1}}{{n}+\mathrm{1}}} −\left[\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\Gamma\left(\frac{\mathrm{1}}{{k}}\right)\right]^{\frac{\mathrm{1}}{{n}}} \\ $$

Question Number 215754 Answers: 1 Comments: 0

Question Number 215755 Answers: 1 Comments: 0

Tow men complet a work in 3hr, three women complet it in 5hr, and five children complet it in 10hr. if we consider among these one man, one woman and one child in how much time they will complet it together?

$${Tow}\:{men}\:{complet}\:{a}\:{work}\:{in}\:\mathrm{3}{hr},\:{three} \\ $$$${women}\:{complet}\:{it}\:{in}\:\mathrm{5}{hr},\:{and}\:{five}\: \\ $$$${children}\:{complet}\:{it}\:{in}\:\mathrm{10}{hr}.\:{if}\:{we}\:{consider} \\ $$$${among}\:{these}\:{one}\:{man},\:{one}\:{woman}\:{and}\:{one} \\ $$$${child}\:{in}\:{how}\:{much}\:{time}\:{they}\:{will} \\ $$$${complet}\:{it}\:{together}? \\ $$

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Determine a, b, c [Lazy problem] J181-2. x^3 −6x^2 +15x−7=(x+a)^3 +bx+c J182-(1) x^3 +ax+2=(x+1)(x^2 +bx+c)

$$\boldsymbol{\mathrm{Determine}}\:\boldsymbol{{a}},\:\boldsymbol{{b}},\:\boldsymbol{{c}}\:\left[\mathrm{Lazy}\:\mathrm{problem}\right] \\ $$$$\mathrm{J181}-\mathrm{2}.\:{x}^{\mathrm{3}} −\mathrm{6}{x}^{\mathrm{2}} +\mathrm{15}{x}−\mathrm{7}=\left({x}+{a}\right)^{\mathrm{3}} +{bx}+{c} \\ $$$$\mathrm{J182}-\left(\mathrm{1}\right)\:{x}^{\mathrm{3}} +{ax}+\mathrm{2}=\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +{bx}+{c}\right) \\ $$

Question Number 215723 Answers: 2 Comments: 0

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