Fun with Loci II

Geometry Level pending

A point P ( x , y ) P(x, y) moves in such a way that its distance from the point A ( 3 , 1 ) A(3, 1) is always three times its distance from the straight line x = − 1 x = -1 . Find the equation of the locus of the moving point P P .

8 x 2 − y 2 + 24 x − 2 y − 1 = 0 8x^2-y^2+24x-2y-1=0 8 x 2 − y 2 + 24 x + 2 y + 1 = 0 8x^2-y^2+24x+2y+1=0 8 x 2 + 9 y 2 − 56 x + 18 y − 89 = 0 8x^2+9y^2-56x+18y-89=0 8 x 2 + 9 y 2 − 56 x − 18 y − 89 = 0 8x^2+9y^2-56x-18y-89=0 8 x 2 + 9 y 2 + 56 x + 18 y + 89 = 0 8x^2+9y^2+56x+18y+89=0 8 x 2 + 9 y 2 − 56 x − 18 y + 89 = 0 8x^2+9y^2-56x-18y+89=0 8 x 2 − y 2 − 24 x − 2 y − 1 = 0 8x^2-y^2-24x-2y-1=0 8 x 2 − y 2 + 24 x + 2 y − 1 = 0 8x^2-y^2+24x+2y-1=0

Tan Chin Cheern by Tan Chin Cheern , 18, Malaysia

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1 solution

Steven Chase
Jul 19, 2019

( x − 3 ) 2 + ( y − 1 ) 2 = 3 ( x + 1 ) ( x − 3 ) 2 + ( y − 1 ) 2 = 9 ( x + 1 ) 2 after some manipulation 8 x 2 − y 2 + 24 x + 2 y − 1 = 0 \sqrt{(x-3)^2 + (y-1)^2} = 3(x+1) \\ (x-3)^2 + (y-1)^2 = 9(x+1)^2 \\ \text{after some manipulation} \\ 8x^2 - y^2 + 24x + 2y - 1 = 0

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