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Locus-of-the-point-z-satisfying-the-equation-iz-1-z-i-2-is-

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Question Number 19738 by Tinkutara last updated on 15/Aug/17
Locus of the point z satisfying the equation ∣iz − 1∣ + ∣z − i∣ = 2 is
Locusofthepointzsatisfyingtheequation∣iz−1∣+∣z−i∣=2is
Commented by math khazana by abdo last updated on 22/Jun/18
let z =x+iy (e) ⇔∣ix−y−1∣ +∣x+i(y−1)∣=2 ⇒(√(x^2 +(y+1)^2 )) +(√(x^2 +(y−1)^2 ))=2 ⇒ (√(x^2 +(y+1)^2 )) =(2−(√(x^2 +(y−1)^2 ))) ⇒ x^2 +(y+1)^2 =4 −4(√(x^2 +(y−1)^2 )) +x^2 +(y−1)^2 ⇒ 4(√(x^2 +(y−1)^2 ))=(y−1)^2 −(y+1)^2 +4 ⇒ 4(√(x^2 +(y−1)^2 ))=y^2 −2y +1 −y^2 −2y −1 +4 =4 −4y ⇒ x^2 +(y−1)^2 =(1−y)^2 ⇒ x=0 ⇒ {z∈C/∣iz−∣+∣z−i∣=2}=iR .
letz=x+iy(e)⇔∣ix−y−1∣+∣x+i(y−1)∣=2⇒x2+(y+1)2+x2+(y−1)2=2⇒x2+(y+1)2=(2−x2+(y−1)2)⇒x2+(y+1)2=4−4x2+(y−1)2+x2+(y−1)2⇒4x2+(y−1)2=(y−1)2−(y+1)2+4⇒4x2+(y−1)2=y2−2y+1−y2−2y−1+4=4−4y⇒x2+(y−1)2=(1−y)2⇒x=0⇒{z∈C/∣iz−∣+∣z−i∣=2}=iR.
Answered by ajfour last updated on 15/Aug/17
⇒ ∣z+i∣+∣z−i∣=2 ellipse with origin as centre. focii : z_1 =−i ; z_2 =i shall try yo obtain equation in the form: (x^2 /a^2 )+(y^2 /b^2 )=1 ... if equation of ellipse be ∣z−z_1 ∣+∣z−z_2 ∣=d with z_1 , and z_2 being focii then 2a=d and b^2 =a^2 −((∣z_1 −z_2 ∣^2 )/4) so eccentricity e=((∣z_1 −z_2 ∣)/d) .
⇒∣z+i∣+∣z−i∣=2ellipsewithoriginascentre.focii:z1=−i;z2=ishalltryyoobtainequationintheform:x2a2+y2b2=1…ifequationofellipsebe∣z−z1∣+∣z−z2∣=dwithz1,andz2beingfociithen2a=dandb2=a2−∣z1−z2∣24soeccentricitye=∣z1−z2∣d.
Commented by ajfour last updated on 16/Aug/17
Here major axis is the Imaginary axis, so 2b=d and a^2 =b^2 −((∣z_1 −z_2 ∣^2 )/4) d=2 , so b=1 z_1 =i , z_2 =−i so a^2 =1−((∣2i∣^2 )/4) =0 hence ellipse with minor axis length zero and major axis length equal to 2. extends from z=−i to z=i , a double line segment. (special case of ellipse).
HeremajoraxisistheImaginaryaxis,so2b=danda2=b2−∣z1−z2∣24d=2,sob=1z1=i,z2=−isoa2=1−∣2i∣24=0henceellipsewithminoraxislengthzeroandmajoraxislengthequalto2.extendsfromz=−itoz=i,adoublelinesegment.(specialcaseofellipse).
Commented by Tinkutara last updated on 16/Aug/17
Sorry, I checked the answer today and it was given a straight line!
Sorry,Icheckedtheanswertodayanditwasgivenastraightline!
Commented by Tinkutara last updated on 16/Aug/17
So we will consider it ellipse or a line?
Sowewillconsideritellipseoraline?
Commented by ajfour last updated on 16/Aug/17
cant say !
cantsay!

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