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Consider the distance formula between two points. The formula is:
`d = sqrt((x_2-x1)^2 + (y_2 - y_1)^2 )`
Then, let `d_1` the distance between (0,0) and (x,y) and
`d_2` the distance between (4,-2) and (x,y).
Since `d_1` and `d_2` are equidistant , set `d_1` ` ` and `d_2` equal to each other.
`d_1 = d_2`
`sqrt((x-0)^2 + (y-0)^2) = sqrt((x-4)^2 + (y - (-2))^2 )`
`sqrt(x^2+y^2) = sqrt((x-4)^2 + (y+2)^2)`
Then, square both sides.
`x^2 + y^2 = (x-4)^2 + (y+2)^2`
Expand right side. Use FOIL method.
`x^2 + y^2 = x^2 - 4x - 4x + 16 + y^2 + 2y + 2y + 4`
`x^2 + y^2 = x^2 - 8x + 16 + y^2 + 4y + 4`
`x^2 + y^2 = x^2 - 8x + y^2 + 4y + 20`
Move all the terms with exponent 2 on one side of the equation.
`x^2 + y^2 - x^2 - y^2 = -8x + 4y + 20`
` 0 = -8x +4y + 20`
Since -8, 4 and 20 are all divisible by 4, then divide both sides by 4 to simplify.
`0 = -2x + y + 5`
Then, we may express the above expression in slope intercept form.
`y = 2x - 5`
Hence, the locus of the point is y = 2x - 5.
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