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Question Number 50898 by ajfour last updated on 21/Dec/18

Commented by ajfour last updated on 21/Dec/18

If length of BP  is maximum  and equal to l, and the two coloured  areas equal, find a and b of ellipse.

IflengthofBPismaximumandequaltol,andthetwocolouredareasequal,findaandbofellipse.

Answered by ajfour last updated on 21/Dec/18

i got  a = l(√(2−(√2)))   ,  b = l(√(3(√2)−4))  .

igota=l22,b=l324.

Commented by mr W last updated on 22/Dec/18

i got the same sir.

igotthesamesir.

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Dec/18

area sky=(1/2)×acosθ×bsinθ+∫_(bsinθ) ^b (a/b)(√(b^2 −y^2 )) dy  =(a/b)[∣(y/2)(√(b^2 −y^2 )) +(b^2 /2)sin^(−1) ((y/b))∣_(bsinθ) ^b ]+(1/4)sin2θ  =(a/b)[((b^2 /2)×(π/2))−(((bsinθ×bcosθ)/2)+(b^2 /2)θ)]+((sin2θ)/4)  =((πab)/4)−((absin2θ)/4)+((abθ)/2)+((sin2θ)/4)  given A_s =A_p =((πab)/8) at θ=θ_0   ((πab)/8)=((πab)/4)−((absin2θ_0 )/4)+((abθ_0 )/2)+((sin2θ_0 )/4)  sin2θ_0 ×(1/4)(ab−1)−((ab)/2)sin^(−1) ((b^2 /(a^2 −b^2 )))=((πab)/8)          length BP=(√((acosθ−0)^2 +(bsinθ+b)^2 )) =s  s_(max) =l given  s^2 =a^2 cos^2 θ+b^2 (1+sinθ)^2   2s×(ds/dθ)=a^2 ×−sin2θ+b^2 ×2(1+sinθ)(cosθ)  2b^2 cosθ+b^2 sin2θ−a^2 sin2θ=0  cosθ[2b^2 +2b^2 sinθ−2a^2 sinθ]=0    cosθ=0  θ=(π/2)←ignored  sinθ_0 =(b^2 /(a^2 −b^2 ))  cosθ_0 =((√((a^2 −b^2 )^2 −b^4 ))/((a^2 −b^2 )))=((√(a^4 −2a^2 b^2 ))/((a^2 −b^2 )))  wait...

areasky=12×acosθ×bsinθ+bsinθbabb2y2dy=ab[y2b2y2+b22sin1(yb)bsinθb]+14sin2θ=ab[(b22×π2)(bsinθ×bcosθ2+b22θ)]+sin2θ4=πab4absin2θ4+abθ2+sin2θ4givenAs=Ap=πab8atθ=θ0πab8=πab4absin2θ04+abθ02+sin2θ04sin2θ0×14(ab1)ab2sin1(b2a2b2)=πab8lengthBP=(acosθ0)2+(bsinθ+b)2=ssmax=lgivens2=a2cos2θ+b2(1+sinθ)22s×dsdθ=a2×sin2θ+b2×2(1+sinθ)(cosθ)2b2cosθ+b2sin2θa2sin2θ=0cosθ[2b2+2b2sinθ2a2sinθ]=0cosθ=0θ=π2ignoredsinθ0=b2a2b2cosθ0=(a2b2)2b4(a2b2)=a42a2b2(a2b2)wait...

Answered by mr W last updated on 22/Dec/18

Commented by OTCHRRE ABDULLAI last updated on 22/Dec/18

My  man you are too good  please you arefrom which country?

Mymanyouaretoogoodpleaseyouarefromwhichcountry?

Commented by mr W last updated on 22/Dec/18

r^2 =((a^2 b^2 )/(a^2 sin^2  θ+b^2 cos^2  θ))  A_(red) =A_(blue) =((πab)/8)  A_(red) =∫_0 ^θ ((r^2 dθ)/2)=((a^2 b^2 )/2)∫_0 ^θ (dθ/(a^2 sin^2  θ+b^2 cos^2  θ))  =((a^2 b^2 )/2)×((tan^(−1) ((a/b)tan θ))/(ab))  =((ab)/2)×tan^(−1) ((a/b)tan θ)=((πab)/8)  ⇒tan^(−1) ((a/b)tan θ)=(π/4)  ⇒(a/b)tan θ=1  ⇒tan θ=(b/a) ⇒ sin θ=(b/(√(a^2 +b^2 ))) and cos θ=(a/(√(a^2 +b^2 )))  P(r,θ)  r^2 =((a^2 b^2 )/(a^2 sin^2  θ+b^2 cos^2  θ))=((a^2 b^2 )/((2a^2 b^2 )/(a^2 +b^2 )))=((a^2 +b^2 )/2)  (x/a^2 )+(y/b^2 )y′=0  ((r cos θ)/a^2 )+((r sin θ)/b^2 )y′=0  ⇒y′(at P)=−(b^2 /(a^2  tan θ))=−(b/a)=−tan α=−tan θ  ⇒α=θ  ϕ=(π/2)−α=(π/2)−θ  r cos θ=l cos ϕ=l sin θ  ⇒r=l tan θ=((lb)/a)  r sin θ=l sin ϕ−b=l cos θ−b  ⇒r=(l/(tan θ))−(b/(sin θ))  ⇒(l/(tan θ))−(b/(sin θ))=l tan θ  ⇒b=l((1/(tan θ))−tan θ)sin θ=l×((a^2 −b^2 )/(ab))×(b/(√(a^2 +b^2 )))  ⇒l=((ab(√(a^2 +b^2 )))/(a^2 −b^2 ))  r^2 =((l^2 b^2 )/a^2 )=(b^2 /a^2 )×((a^2 b^2 (a^2 +b^2 ))/((a^2 −b^2 )^2 ))=((b^4 (a^2 +b^2 ))/((a^2 −b^2 )^2 ))=((a^2 +b^2 )/2)  ⇒(b^4 /((a^2 −b^2 )^2 ))=(1/2)  ((√2)b^2 )^2 −(a^2 −b^2 )^2 =0  [((√2)+1)b^2 −a^2 ][((√2)−1)b^2 +a^2 ]=0  ⇒a^2 =((√2)+1)b^2   ⇒l=((ab(√(a^2 +b^2 )))/(a^2 −b^2 ))=((ab(√((2+(√2))b^2 )))/((√2)b^2 ))=a(√((2+(√2))/2))  ⇒a=(√(2/(2+(√2)))) l=(√(2−(√2))) l  ⇒b=(a/(√((√2)+1)))=(√((2−(√2))/((√2)+1))) l=(√(3(√2)−4)) l

r2=a2b2a2sin2θ+b2cos2θAred=Ablue=πab8Ared=0θr2dθ2=a2b220θdθa2sin2θ+b2cos2θ=a2b22×tan1(abtanθ)ab=ab2×tan1(abtanθ)=πab8tan1(abtanθ)=π4abtanθ=1tanθ=basinθ=ba2+b2andcosθ=aa2+b2P(r,θ)r2=a2b2a2sin2θ+b2cos2θ=a2b22a2b2a2+b2=a2+b22xa2+yb2y=0rcosθa2+rsinθb2y=0y(atP)=b2a2tanθ=ba=tanα=tanθα=θφ=π2α=π2θrcosθ=lcosφ=lsinθr=ltanθ=lbarsinθ=lsinφb=lcosθbr=ltanθbsinθltanθbsinθ=ltanθb=l(1tanθtanθ)sinθ=l×a2b2ab×ba2+b2l=aba2+b2a2b2r2=l2b2a2=b2a2×a2b2(a2+b2)(a2b2)2=b4(a2+b2)(a2b2)2=a2+b22b4(a2b2)2=12(2b2)2(a2b2)2=0[(2+1)b2a2][(21)b2+a2]=0a2=(2+1)b2l=aba2+b2a2b2=ab(2+2)b22b2=a2+22a=22+2l=22lb=a2+1=222+1l=324l

Commented by ajfour last updated on 22/Dec/18

WOW !

WOW!

Commented by mr W last updated on 22/Dec/18

Commented by mr W last updated on 22/Dec/18

diagram shows the case l=5.

diagramshowsthecasel=5.

Commented by ajfour last updated on 22/Dec/18

Too Good Sir, Thanks for confirming.

TooGoodSir,Thanksforconfirming.

Commented by OTCHRRE ABDULLAI last updated on 23/Dec/18

You are world best my man   if mathematics is football you will   be messi or Ronaldo i really like   you infact you are now my best   friend

YouareworldbestmymanifmathematicsisfootballyouwillbemessiorRonaldoireallylikeyouinfactyouarenowmybestfriend

Commented by peter frank last updated on 22/Dec/18

and physics too

andphysicstoo

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