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 Question from srishti, a student: A point P moves such that the difference between its distance from the origin and from the axis of x is always a constant c . What is the locus of the point?

Hi Srishti,

Suppose $(x, y)$ is a point on the locus then its distance from the origin is $\sqrt{x^2 + y^2}.$

Suppose that $y \geq 0$ then the distance from $(x, y)$ to the X-axis is $y$ and hence for $(x, y)$ to be on the locus you must have

$\sqrt{x^2 + y^2} - y = c.$

Simplify and solve for $y.$

Now suppose that $y < 0$ then the distance from $(x, y)$ to the X-axis is $-y.$ Write an expression analogous to the expression above, simplify and solve for $y.$

Thus the locus of $(x, y)$ is given in two parts, one for $y \geq 0$ and the other for $y < 0.$

Penny

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