Distance between co-ordinates in a plane

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A and B are two points on a co-ordinate plane. All the points, in the same plane as A and B, whose distance from B is twice that from A lie on



  • a a straight line intersecting AB at a point O such that 2AO = BO.


  • b a circle with center at a point O on AB such that AO = 2BO.


  • c a circle with center at a point O on AB extended such that 4AO = BO..


  • d None of these.




geometry circle coordinate-systems plane-geometry




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    Lookup Apollonius' circles.
    – dxiv
    Aug 1 at 4:25














up vote
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down vote

favorite
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A and B are two points on a co-ordinate plane. All the points, in the same plane as A and B, whose distance from B is twice that from A lie on



  • a a straight line intersecting AB at a point O such that 2AO = BO.


  • b a circle with center at a point O on AB such that AO = 2BO.


  • c a circle with center at a point O on AB extended such that 4AO = BO..


  • d None of these.




geometry circle coordinate-systems plane-geometry




share|cite|improve this question















  • 2




    Lookup Apollonius' circles.
    – dxiv
    Aug 1 at 4:25












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





A and B are two points on a co-ordinate plane. All the points, in the same plane as A and B, whose distance from B is twice that from A lie on



  • a a straight line intersecting AB at a point O such that 2AO = BO.


  • b a circle with center at a point O on AB such that AO = 2BO.


  • c a circle with center at a point O on AB extended such that 4AO = BO..


  • d None of these.




geometry circle coordinate-systems plane-geometry




share|cite|improve this question











A and B are two points on a co-ordinate plane. All the points, in the same plane as A and B, whose distance from B is twice that from A lie on



  • a a straight line intersecting AB at a point O such that 2AO = BO.


  • b a circle with center at a point O on AB such that AO = 2BO.


  • c a circle with center at a point O on AB extended such that 4AO = BO..


  • d None of these.





geometry circle coordinate-systems plane-geometry





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asked Aug 1 at 4:20









Krishna Barri

1083




1083







  • 2




    Lookup Apollonius' circles.
    – dxiv
    Aug 1 at 4:25












  • 2




    Lookup Apollonius' circles.
    – dxiv
    Aug 1 at 4:25







2




2




Lookup Apollonius' circles.
– dxiv
Aug 1 at 4:25




Lookup Apollonius' circles.
– dxiv
Aug 1 at 4:25










1 Answer
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Suppose that $A(0,0)$, $B(a, 0)$ and $C(x,y)$.



$$2CA=CB implies$$



$$2sqrtx^2+y^2=sqrt(x-a)^2+y^2$$



$$4(x^2+y^2)=(x-a)^2+y^2$$



$$4x^2+4y^2=x^2-2ax+a^2+y^2$$



$$3x^2+2ax+3y^2=a^2$$



$$x^2+frac23ax+y^2=frac13a^2$$



$$(x+frac13a)^2-frac19a^2+y^2=frac13a^2$$



$$(x+frac13a)^2+y^2=frac49a^2$$



...which is a circle with center $O(-frac13a,0)$.



It means that $OA=frac13a$, $OB=frac43a$ or $OB=4OA$.



So the right answer is answer (c).






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    1 Answer
    1






    active

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    active

    oldest

    votes








    up vote
    0
    down vote



    accepted










    Suppose that $A(0,0)$, $B(a, 0)$ and $C(x,y)$.



    $$2CA=CB implies$$



    $$2sqrtx^2+y^2=sqrt(x-a)^2+y^2$$



    $$4(x^2+y^2)=(x-a)^2+y^2$$



    $$4x^2+4y^2=x^2-2ax+a^2+y^2$$



    $$3x^2+2ax+3y^2=a^2$$



    $$x^2+frac23ax+y^2=frac13a^2$$



    $$(x+frac13a)^2-frac19a^2+y^2=frac13a^2$$



    $$(x+frac13a)^2+y^2=frac49a^2$$



    ...which is a circle with center $O(-frac13a,0)$.



    It means that $OA=frac13a$, $OB=frac43a$ or $OB=4OA$.



    So the right answer is answer (c).






    share|cite|improve this answer

























      up vote
      0
      down vote



      accepted










      Suppose that $A(0,0)$, $B(a, 0)$ and $C(x,y)$.



      $$2CA=CB implies$$



      $$2sqrtx^2+y^2=sqrt(x-a)^2+y^2$$



      $$4(x^2+y^2)=(x-a)^2+y^2$$



      $$4x^2+4y^2=x^2-2ax+a^2+y^2$$



      $$3x^2+2ax+3y^2=a^2$$



      $$x^2+frac23ax+y^2=frac13a^2$$



      $$(x+frac13a)^2-frac19a^2+y^2=frac13a^2$$



      $$(x+frac13a)^2+y^2=frac49a^2$$



      ...which is a circle with center $O(-frac13a,0)$.



      It means that $OA=frac13a$, $OB=frac43a$ or $OB=4OA$.



      So the right answer is answer (c).






      share|cite|improve this answer























        up vote
        0
        down vote



        accepted







        up vote
        0
        down vote



        accepted






        Suppose that $A(0,0)$, $B(a, 0)$ and $C(x,y)$.



        $$2CA=CB implies$$



        $$2sqrtx^2+y^2=sqrt(x-a)^2+y^2$$



        $$4(x^2+y^2)=(x-a)^2+y^2$$



        $$4x^2+4y^2=x^2-2ax+a^2+y^2$$



        $$3x^2+2ax+3y^2=a^2$$



        $$x^2+frac23ax+y^2=frac13a^2$$



        $$(x+frac13a)^2-frac19a^2+y^2=frac13a^2$$



        $$(x+frac13a)^2+y^2=frac49a^2$$



        ...which is a circle with center $O(-frac13a,0)$.



        It means that $OA=frac13a$, $OB=frac43a$ or $OB=4OA$.



        So the right answer is answer (c).






        share|cite|improve this answer













        Suppose that $A(0,0)$, $B(a, 0)$ and $C(x,y)$.



        $$2CA=CB implies$$



        $$2sqrtx^2+y^2=sqrt(x-a)^2+y^2$$



        $$4(x^2+y^2)=(x-a)^2+y^2$$



        $$4x^2+4y^2=x^2-2ax+a^2+y^2$$



        $$3x^2+2ax+3y^2=a^2$$



        $$x^2+frac23ax+y^2=frac13a^2$$



        $$(x+frac13a)^2-frac19a^2+y^2=frac13a^2$$



        $$(x+frac13a)^2+y^2=frac49a^2$$



        ...which is a circle with center $O(-frac13a,0)$.



        It means that $OA=frac13a$, $OB=frac43a$ or $OB=4OA$.



        So the right answer is answer (c).







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered Aug 1 at 8:50









        Oldboy

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