Straight Lines #2

Geometry Level 2

To which point the origin should be shifted so that the terms linear in x x and y y in the equation x 2 + y 2 − 2 x + 6 y + 8 = 0 x^2+y^2-2x+6y+8=0 are eliminated?

( − 2 , 3 ) (-2,3) ( − 1 , 3 ) (-1,3) ( 2 , − 2 ) (2,-2) ( 1 , − 3 ) (1,-3)

Aakhyat Singh by Aakhyat Singh , 20, India

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1 solution

Chew-Seong Cheong
Aug 28, 2017

x 2 + y 2 − 2 x + 6 y + 8 = 0 x 2 − 2 x + 1 + y 2 + 6 y + 9 − 2 = 0 ( x − 1 ) 2 + ( y + 3 ) 2 − 2 = 0 \begin{aligned} x^2 + y^2 - 2x + 6y + 8 & = 0 \\ x^2 - 2x + 1 + y^2 + 6y + 9 - 2 & = 0 \\ (x \ {\color{#3D99F6}-\ 1})^2 + (y\ {\color{#3D99F6}+\ 3})^2 - 2 & = 0 \end{aligned}

The origin has to be shifted to ( 1 , − 3 ) \boxed{(1,-3)} so that terms linear to x x and y y are eliminated.

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