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Question Number 45565    Answers: 2   Comments: 0

If ax^2 +by^2 +2hxy+2gx+2fy+c=0 be the equation of an ellipse, find coordinates of center of ellipse. Q.45506 (another solution)

$${If}\:\:\boldsymbol{{ax}}^{\mathrm{2}} +\boldsymbol{{by}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{{hxy}}+\mathrm{2}\boldsymbol{{gx}}+\mathrm{2}\boldsymbol{{fy}}+\boldsymbol{{c}}=\mathrm{0} \\ $$$${be}\:{the}\:{equation}\:{of}\:{an}\:{ellipse},\:{find} \\ $$$${coordinates}\:{of}\:{center}\:{of}\:{ellipse}. \\ $$$${Q}.\mathrm{45506}\:\:\left({another}\:{solution}\right) \\ $$

Question Number 45563    Answers: 0   Comments: 0

Question Number 45561    Answers: 0   Comments: 2

Question Number 45555    Answers: 1   Comments: 1

Prove that points (4,−1,3) & (5,−1,4) lies on same side of the plane x+y+z=7.

$${Prove}\:{that}\:{points}\:\left(\mathrm{4},−\mathrm{1},\mathrm{3}\right)\:\&\:\left(\mathrm{5},−\mathrm{1},\mathrm{4}\right) \\ $$$${lies}\:{on}\:{same}\:{side}\:{of}\:{the}\:{plane}\:{x}+{y}+{z}=\mathrm{7}. \\ $$

Question Number 45546    Answers: 2   Comments: 1

Simplify: (((√5) + 2))^(1/3) + (((√5) − 2))^(1/3)

$$\mathrm{Simplify}:\:\:\:\:\:\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}\:\:+\:\:\mathrm{2}}\:\:\:+\:\:\:\:\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}\:\:−\:\mathrm{2}}\:\: \\ $$

Question Number 45539    Answers: 1   Comments: 2

Question Number 45543    Answers: 0   Comments: 0

Question Number 45527    Answers: 0   Comments: 2

Question Number 45520    Answers: 0   Comments: 1

let a>0 and b>0 calculate ∫ (√(acos^2 θ +bsin^2 θ))dπ 2) find ∫_(π/4) ^(π/2) (√(2cos^2 θ +3 sin^2 θ))dθ .

$${let}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:{calculate}\:\int\:\sqrt{{acos}^{\mathrm{2}} \theta\:+{bsin}^{\mathrm{2}} \theta}{d}\pi \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{2}{cos}^{\mathrm{2}} \theta\:+\mathrm{3}\:{sin}^{\mathrm{2}} \theta}{d}\theta\:. \\ $$

Question Number 45519    Answers: 0   Comments: 2

find ∫ (√(2+tan^2 θ))dθ

$${find}\:\:\int\:\sqrt{\mathrm{2}+{tan}^{\mathrm{2}} \theta}{d}\theta \\ $$

Question Number 45518    Answers: 2   Comments: 1

calculate Σ_(n=1) ^∞ ((cos(nθ))/n^2 ) and Σ_(n=1) ^∞ ((sin(nθ))/n^2 )

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{cos}\left({n}\theta\right)}{{n}^{\mathrm{2}} }\:\:{and}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{sin}\left({n}\theta\right)}{{n}^{\mathrm{2}} } \\ $$

Question Number 45517    Answers: 0   Comments: 0

find S(x)=Σ_(n=1) ^∞ (x^n /n^2 ) with ∣x∣<1 .

$${find}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{x}^{{n}} }{{n}^{\mathrm{2}} }\:\:{with}\:\mid{x}\mid<\mathrm{1}\:\:. \\ $$

Question Number 45514    Answers: 0   Comments: 1

Question Number 45512    Answers: 0   Comments: 11

Calculate: (((2^4 + (1/4)) (4^4 + (1/4))(6^4 + (1/4))(8^4 + (1/4))(10^4 + (1/4))(12^4 + (1/4)))/((1^4 + (1/4))(3^4 + (1/4)) (5^4 + (1/4)) (7^4 + (1/4)) (9^4 + (1/4))(11^4 + (1/4))))

$$\mathrm{Calculate}:\:\:\:\frac{\left(\mathrm{2}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\:\left(\mathrm{4}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{6}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{8}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{10}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{12}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)}{\left(\mathrm{1}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{3}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\:\left(\mathrm{5}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\:\left(\mathrm{7}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\:\left(\mathrm{9}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{11}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)} \\ $$

Question Number 45511    Answers: 0   Comments: 0

In a triangle ABC , ∠ ABC = 30^0 , and AC = 10. A circle is drawn to circumscribe the triangle . Find the radius of the circle.

$$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{ABC}\:,\:\:\angle\:\mathrm{ABC}\:=\:\mathrm{30}^{\mathrm{0}} \:,\:\:\mathrm{and}\:\:\mathrm{AC}\:=\:\mathrm{10}.\:\mathrm{A}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{to} \\ $$$$\mathrm{circumscribe}\:\mathrm{the}\:\mathrm{triangle}\:.\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}. \\ $$

Question Number 45506    Answers: 1   Comments: 0

If ax^2 +by^2 +2hxy+2gx+2fy+c=0 be the equation of an ellipse, find coordinates of its centre.

$${If}\:\:\:{ax}^{\mathrm{2}} +{by}^{\mathrm{2}} +\mathrm{2}{hxy}+\mathrm{2}{gx}+\mathrm{2}{fy}+{c}=\mathrm{0} \\ $$$${be}\:{the}\:{equation}\:{of}\:{an}\:{ellipse},\:{find} \\ $$$${coordinates}\:{of}\:{its}\:{centre}. \\ $$

Question Number 45500    Answers: 1   Comments: 0

Question Number 45498    Answers: 1   Comments: 3

Question Number 45495    Answers: 1   Comments: 0

Question Number 45494    Answers: 0   Comments: 0

Question Number 45477    Answers: 0   Comments: 4

Question Number 45482    Answers: 1   Comments: 0

Question Number 45464    Answers: 1   Comments: 3

Differentiate with respect to x arctan(((a^2 +x^2 )/(a^2 −x^2 )))

$$\boldsymbol{\mathrm{D}}\mathrm{ifferentiate}\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{arctan}}\left(\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right) \\ $$

Question Number 45456    Answers: 1   Comments: 5

Question Number 45451    Answers: 1   Comments: 0

Question Number 45450    Answers: 1   Comments: 1

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