A point A moves so that its distance to (0, 0) is always equal to twice its distance to the point (3, 0). The equation of the circle is?

1 Answer
CW
Nov 19, 2017

equation of the circle : #(x-4)^2+y^2=4#

Explanation:

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Let the moving point be #P(x,y), A=(0,0), and B=(3,0)#, as shown in the figure.
#AP^2=x^2+y^2#,
#BP^2=(x-3)^2+y^2=x^2-6x+9+y^2#,
Given #AP=2BP, => AP^2=4BP^2#
#=> x^2+y^2=4(x^2-6x+9+y^2)#
#=> x^2+y^2=4x^2-24x+36+4y^2#
#=> 3x^2-24x+36+3y^2=0#
#=> x^2-8x+12+y^2=0#
#=> (x-4)^2-16+12+y^2=0#
#=> color(red)((x-4)^2+y^2=4)#
Hence, the locus is a circle centered at #(4,0)# and the radius #=2#

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