Introduction:

To find the equation of the locus, we need to understand the given condition and derive the equation based on it. Let's break down the problem step by step.

Understanding the given condition:

The given condition states that the distance of a point (x, y) from the origin (0, 0) is twice its distance from the y-axis. Mathematically, we can represent this condition as:

√(x^2 + y^2) = 2 * |x|

Here, the left-hand side of the equation represents the distance of the point (x, y) from the origin, and the right-hand side represents twice the distance of the point (x, y) from the y-axis.

Deriving the equation:

To find the equation of the locus, we need to eliminate the absolute value sign from the equation. We can do this by considering two cases:

Case 1: x ≥ 0
In this case, the equation |x| simplifies to just x. Substituting this in the given condition, we get:

√(x^2 + y^2) = 2x

Squaring both sides of the equation, we get:

x^2 + y^2 = 4x^2

Rearranging the terms, we get:

3x^2 - y^2 = 0

Case 2: x < />
In this case, the equation |x| simplifies to -x. Substituting this in the given condition, we get:

√(x^2 + y^2) = -2x

Squaring both sides of the equation, we get:

x^2 + y^2 = 4x^2

Rearranging the terms, we get:

3x^2 - y^2 = 0

Combining both cases:

Since the result is the same in both cases, we can combine them into a single equation:

3x^2 - y^2 = 0

Conclusion:

The equation of the locus is 3x^2 - y^2 = 0, which represents all the points that satisfy the condition mentioned in the problem. The locus is a hyperbola with the x-axis as its transverse axis, passing through the origin.

y square =0