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AlgebraQuestion and Answers: Page 268

Question Number 83590 Answers: 0 Comments: 3

transform the ellipse (x^2 /a^2 )+(y^2 /b^2 )=1 to the polar equation r= ((a(1−e^2 ))/(1+ecosθ)) a: semimajor axis e: eccentricity

$${transform}\:{the}\:{ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1}\:{to} \\ $$$${the}\:{polar}\:{equation}\:{r}=\:\frac{{a}\left(\mathrm{1}−{e}^{\mathrm{2}} \right)}{\mathrm{1}+{ecos}\theta} \\ $$$${a}:\:{semimajor}\:{axis} \\ $$$${e}:\:{eccentricity} \\ $$

Question Number 83570 Answers: 2 Comments: 3

Find the locus of a point which moves such that its distance from the line y = 4 is a constant k.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{a}\:\mathrm{point}\:\mathrm{which}\:\mathrm{moves}\:\mathrm{such}\:\mathrm{that}\:\mathrm{its} \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{line}\:\:\:\mathrm{y}\:\:=\:\:\mathrm{4}\:\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\:\:\mathrm{k}. \\ $$

Question Number 83554 Answers: 1 Comments: 0

Question Number 83543 Answers: 1 Comments: 2

Question Number 83542 Answers: 2 Comments: 0

Question Number 83621 Answers: 2 Comments: 1

∣ x+(1/x)∣ < 4 find the solution

$$\mid\:{x}+\frac{\mathrm{1}}{{x}}\mid\:<\:\mathrm{4}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution} \\ $$

Question Number 83512 Answers: 1 Comments: 0

find the solution ((4x^2 )/((1−(√(2x+1)))^2 )) < 2x+9

$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\: \\ $$$$\frac{\mathrm{4x}^{\mathrm{2}} }{\left(\mathrm{1}−\sqrt{\mathrm{2x}+\mathrm{1}}\right)^{\mathrm{2}} }\:<\:\mathrm{2x}+\mathrm{9} \\ $$

Question Number 83477 Answers: 0 Comments: 1

a^b +b^a =1 a=? , b=? a≠b≠0

$${a}^{{b}} +{b}^{{a}} =\mathrm{1}\:\:\:{a}=?\:,\:{b}=? \\ $$$${a}\neq{b}\neq\mathrm{0} \\ $$

Question Number 83473 Answers: 1 Comments: 0

solve in R sin(πln(x))+cos(πln(x))=1

$${solve}\:{in}\:{R} \\ $$$${sin}\left(\pi{ln}\left({x}\right)\right)+{cos}\left(\pi{ln}\left({x}\right)\right)=\mathrm{1} \\ $$

Question Number 83378 Answers: 0 Comments: 2

x+y+z=1 x^2 +y^2 +z^2 =2 x^3 +y^3 +z^3 =3 find x^4 +y^4 +z^4 =?

$${x}+{y}+{z}=\mathrm{1} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} =\mathrm{2} \\ $$$${x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} =\mathrm{3} \\ $$$${find} \\ $$$${x}^{\mathrm{4}} +{y}^{\mathrm{4}} +{z}^{\mathrm{4}} =? \\ $$

Question Number 83373 Answers: 0 Comments: 4

Question Number 83372 Answers: 0 Comments: 1

Question Number 83370 Answers: 0 Comments: 3

Question Number 83341 Answers: 1 Comments: 1

Find the maximum and minimum of the expression 𝚺_(i=1) ^n a_i x_i with 𝚺_(i=1) ^n (x_i −b_i )^2 =c^2 , where a_i , b_i and c are constants. (extracted and modified from Q83331)

$${Find}\:{the}\:{maximum}\:{and}\:{minimum} \\ $$$${of}\:{the}\:{expression}\:\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}{a}}_{\boldsymbol{{i}}} \boldsymbol{{x}}_{\boldsymbol{{i}}} \:{with} \\ $$$$\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\boldsymbol{\sum}}}\left(\boldsymbol{{x}}_{\boldsymbol{{i}}} −\boldsymbol{{b}}_{\boldsymbol{{i}}} \right)^{\mathrm{2}} =\boldsymbol{{c}}^{\mathrm{2}} ,\:{where}\:\boldsymbol{{a}}_{\boldsymbol{{i}}} ,\:\boldsymbol{{b}}_{\boldsymbol{{i}}} \:{and}\:\boldsymbol{{c}}\:{are} \\ $$$${constants}. \\ $$$$ \\ $$$$\left({extracted}\:{and}\:{modified}\:{from}\:{Q}\mathrm{83331}\right) \\ $$

Question Number 83264 Answers: 1 Comments: 2

Question Number 83262 Answers: 1 Comments: 0

Question Number 83229 Answers: 0 Comments: 4

Question Number 83050 Answers: 0 Comments: 1

Question Number 83049 Answers: 0 Comments: 0

Question Number 82991 Answers: 1 Comments: 2

Question Number 82952 Answers: 1 Comments: 0

Show that: y + (√(y^2 − 1)) ≥ 1 and 0 < y − (√(y^2 − 1)) ≤ 1 if y ≥ 1

$$\mathrm{Show}\:\mathrm{that}:\:\:\:\:\:\:\mathrm{y}\:\:+\:\:\sqrt{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{1}}\:\:\:\geqslant\:\:\mathrm{1}\:\:\:\:\:\mathrm{and}\:\:\:\:\mathrm{0}\:\:<\:\:\mathrm{y}\:\:−\:\:\sqrt{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{1}}\:\:\leqslant\:\:\mathrm{1} \\ $$$$\mathrm{if}\:\:\mathrm{y}\:\:\geqslant\:\mathrm{1} \\ $$

Question Number 82897 Answers: 1 Comments: 2

Question Number 82881 Answers: 1 Comments: 2

(√(√(...(√(6561))))) = 3^8^x (60 times) find x

$$ \\ $$$$\sqrt{\sqrt{...\sqrt{\mathrm{6561}}}}\:=\:\mathrm{3}^{\mathrm{8}^{\mathrm{x}} } \:\left(\mathrm{60}\:\mathrm{times}\right) \\ $$$$\mathrm{find}\:\mathrm{x} \\ $$

Question Number 82873 Answers: 1 Comments: 2

Question Number 82843 Answers: 1 Comments: 0

show that (((1+(√3) i)^4 (1+i)^8 )/((cos100°−i sin100)^3 ))=−256

$${show}\:{that} \\ $$$$\frac{\left(\mathrm{1}+\sqrt{\mathrm{3}}\:{i}\right)^{\mathrm{4}} \left(\mathrm{1}+{i}\right)^{\mathrm{8}} }{\left({cos}\mathrm{100}°−{i}\:{sin}\mathrm{100}\right)^{\mathrm{3}} }=−\mathrm{256} \\ $$

Question Number 82877 Answers: 1 Comments: 1

1)find xy∈R 2)find x,y∈Z (x+2yi)^6 =8i

$$\left.\mathrm{1}\right){find}\:{xy}\in{R} \\ $$$$\left.\mathrm{2}\right){find}\:{x},{y}\in{Z} \\ $$$$\left({x}+\mathrm{2}{yi}\right)^{\mathrm{6}} =\mathrm{8}{i} \\ $$

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