Menu Close

Find-the-locus-of-the-points-represented-by-the-complex-number-z-such-that-2-z-3-z-6i-

Tinku Tara 0 Comments
Pin




Question Number 83991 by Rio Michael last updated on 08/Mar/20
Find the locus of the points represented by the complex number ,z, such that 2∣z−3∣ = ∣z−6i∣
Findthelocusofthepointsrepresentedbythecomplexnumber,z,suchthat2∣z−3∣=∣z−6i∣
Commented by mathmax by abdo last updated on 08/Mar/20
let z=x+iy (e)⇔2∣x+iy−3∣=∣x+iy−6i∣ ⇒ 2(√((x−3)^2 +y^2 ))=(√(x^2 +(y−6)^2 )) ⇒4{(x−3)^2 +y^2 }=x^2 +(y−6)^2 ⇒ 4(x^2 −6x +9 +y^2 )=x^2 +y^2 −12y +36 ⇒ 4x^2 −24x +36 +4y^2 =x^2 +y^2 −12y +36 ⇒ 4x^2 −24x+4y^2 −x^2 −y^2 +12y =0 ⇒ 3x^2 +3y^2 −24x +12y =0 ⇒x^2 +y^2 −8x +4y =0 ⇒ x^2 −8x +16 −16 +y^2 +4y +4−4 =0 ⇒ (x−4)^2 +(y+2)^2 =(2(√5))^2 so the locus is circle with centre w(4,−2) and radius r=2(√5)
letz=x+iy(e)⇔2∣x+iy−3∣=∣x+iy−6i∣⇒2(x−3)2+y2=x2+(y−6)2⇒4{(x−3)2+y2}=x2+(y−6)2⇒4(x2−6x+9+y2)=x2+y2−12y+36⇒4x2−24x+36+4y2=x2+y2−12y+36⇒4x2−24x+4y2−x2−y2+12y=0⇒3x2+3y2−24x+12y=0⇒x2+y2−8x+4y=0⇒x2−8x+16−16+y2+4y+4−4=0⇒(x−4)2+(y+2)2=(25)2sothelocusiscirclewithcentrew(4,−2)andradiusr=25

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *