Find the equation of the locus of a point that it is equidistant from (-2,4) and the y-axis.

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Let a point has coordinates `x` and `y.` Then the distance between this point and the given point (-2, 4) is

`sqrt((x+2)^2+(y-4)^2).`

The distance from (x, y) to the y-axis is `|x|.`

The equation is

`sqrt((x+2)^2+(y-4)^2)=|x|,`

but we can (and have to) simplify it. Both sides are non-negative, therefore we can square both sides and obtain an equivalent equation:

`(x+2)^2+(y-4)^2=|x|^2=x^2.`

The left side is equal to

`x^2+4x+4+(y-4)^2,`

so the equation may be written as

`4x+4+(y-4)^2=0,` or `x=-1-(y-4)^2/4.`

This is the equation of a parabola. Its focus is (-2, 4) and its axis of symmetry is y=4. Also look at the picture attached or at this link.

axis of symmetry
axis of symmetry

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