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The tangent of a parabola y^2 =4ax at the point P (ap^2 , 2ap) intersects the line x+a=0 at T . (i) If M is the midpoint of PT , find the coordinates of M in terms of a and p. (ii) Prove that the equation of locus of M is y^2 (2x+a)=a(3x+a)^2

$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{of}\:\mathrm{a}\:\mathrm{parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{ax}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point} \\ $$$${P}\:\left({ap}^{\mathrm{2}} ,\:\mathrm{2}{ap}\right)\:\mathrm{intersects}\:\mathrm{the}\:\mathrm{line}\:{x}+{a}=\mathrm{0}\:\mathrm{at}\:{T}\:. \\ $$$$\left(\mathrm{i}\right)\:\mathrm{If}\:{M}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:{PT}\:,\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\:\:\:\:\:\:\mathrm{coordinates}\:\mathrm{of}\:{M}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{a}\:\mathrm{and}\:{p}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{locus}\:\mathrm{of}\:{M}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:{y}^{\mathrm{2}} \left(\mathrm{2}{x}+{a}\right)={a}\left(\mathrm{3}{x}+{a}\right)^{\mathrm{2}} \\ $$

Question Number 162249 Answers: 1 Comments: 0

Question Number 162243 Answers: 2 Comments: 0

∫_( 0) ^( 1) ((((ln x)^4 )/( (√x) ))) dx =?

$$\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\left(\frac{\left(\mathrm{ln}\:{x}\right)^{\mathrm{4}} }{\:\sqrt{{x}}\:}\right)\:{dx}\:=? \\ $$

Question Number 162240 Answers: 1 Comments: 0

∫((2x−5)/(x^2 +4x+5))dx

$$\int\frac{\mathrm{2}\boldsymbol{{x}}−\mathrm{5}}{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{4}\boldsymbol{{x}}+\mathrm{5}}\boldsymbol{{dx}} \\ $$

Question Number 162238 Answers: 1 Comments: 0

∫_0 ^1 (1/(x^7 +1))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{7}} +\mathrm{1}}\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 162234 Answers: 0 Comments: 0

Σ_(p=0) ^n (−1)^p C_p ^(1/2) (−1)^(n−p) C_(n−p) ^(1/2) = ???? Please Help me(Aidez moi)

$$ \\ $$$$\:\:\:\:\:\:\underset{{p}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{p}} {C}_{{p}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \left(−\mathrm{1}\right)^{{n}−{p}} {C}_{{n}−{p}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:=\:\:\:???? \\ $$$$\:\:\:\:\:\:\:{Please}\:{Help}\:{me}\left({Aidez}\:{moi}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 162229 Answers: 1 Comments: 0

Question Number 162287 Answers: 0 Comments: 4

lim_(n→∞) ∫_0 ^1 ((x^n +(1−x)^n ))^(1/n) dx=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt[{\mathrm{n}}]{\mathrm{x}^{\mathrm{n}} +\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{n}} }\mathrm{dx}=? \\ $$

Question Number 162219 Answers: 2 Comments: 0

Ω=∫_0 ^1 ((log(1+x^7 ))/(1+x^7 ))dx=?

$$\Omega=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{log}\left(\mathrm{1}+{x}^{\mathrm{7}} \right)}{\mathrm{1}+{x}^{\mathrm{7}} }{dx}=? \\ $$

Question Number 162209 Answers: 1 Comments: 2

Question Number 162190 Answers: 1 Comments: 0

⌊ ((3^2 +1)/(3^2 −1)) + ((3^3 +1)/(3^3 −1)) + ((3^4 +1)/(3^4 −1)) + …+ ((3^(2017) +1)/(3^(2017) −1)) ⌋ = ?

$$\lfloor\:\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{1}}{\mathrm{3}^{\mathrm{2}} −\mathrm{1}}\:+\:\frac{\mathrm{3}^{\mathrm{3}} +\mathrm{1}}{\mathrm{3}^{\mathrm{3}} −\mathrm{1}}\:+\:\frac{\mathrm{3}^{\mathrm{4}} +\mathrm{1}}{\mathrm{3}^{\mathrm{4}} −\mathrm{1}}\:+\:\ldots+\:\frac{\mathrm{3}^{\mathrm{2017}} +\mathrm{1}}{\mathrm{3}^{\mathrm{2017}} −\mathrm{1}}\:\rfloor\:=\:\:? \\ $$

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