Understanding the Locus of a Point Based on Its Distance from the Y-axis and Origin

Let's consider a point P with coordinates (x, y). We are interested in finding the set of points such that the distance of these points from the y-axis is half the distance from the origin. This problem can be solved through a step-by-step process, as detailed below.

Making Sense of the Given Conditions

For the point P with coordinates (x, y), its distance from the y-axis is simply the x-coordinate, denoted as x. The distance from the origin is given by the Euclidean distance, which is (sqrt{x^2 y^2}). The condition provided states that the distance from the y-axis is half the distance from the origin, which can be mathematically represented as:

Equation Representation:

x (frac{1}{2}sqrt{x^2 y^2})

Solving the Equation

To solve for the locus of the point P, we need to isolate x and y in the given equation. First, we will square both sides of the equation to eliminate the square root:

Step 1: Squaring Both Sides

(x^2 left(frac{1}{2}sqrt{x^2 y^2}right)^2)

This simplifies to:

(x^2 frac{1}{4}(x^2 y^2))

Multiplying both sides of the equation by 4 to clear the fraction:

(4x^2 x^2 y^2)

Subtracting (x^2) from both sides of the equation:

(3x^2 y^2)

Final Equation for the Locus:

The locus of the point P, which satisfies the given condition, can be written as:

(y pmsqrt{3}x)

This represents two lines:

(y sqrt{3}x) (y -sqrt{3}x)

Geometric Interpretation

These equations indicate that the locus of such points forms two lines that intersect at the origin. The first line, (y sqrt{3}x), has a slope of (sqrt{3}) and the second line, (y -sqrt{3}x), has a slope of (-sqrt{3}). The slope (sqrt{3}) is equivalent to the tangent of 60 degrees, and thus, the lines form an angle of 60 degrees with the x-axis.

Therefore, the equation of the locus of the point P is:

(y sqrt{3}x)

(y -sqrt{3}x)

Readers can visualize these lines as symmetrical about the origin, forming a 'X' shape when plotted on a coordinate plane.

Conclusion

This locus provides a clear geometric representation of all points that are at a distance from the y-axis that is half their distance from the origin. Such problems are common in coordinate geometry and help in understanding the relationship between distances in the coordinate plane.

Have a Good One!

Subhamoy Chattaraj