To find the equation of the locus of a point which moves so that its distance from the Y-axis is half of its distance from the origin, follow these steps:
1. Coordinates and Distance Relationships:
- Let the coordinates of the moving point be [tex]\((x, y)\)[/tex].
- The distance of the point [tex]\((x, y)\)[/tex] from the Y-axis is [tex]\(|x|\)[/tex]. This is because the Y-axis corresponds to [tex]\(x = 0\)[/tex], so the distance from any point [tex]\((x, y)\)[/tex] to the Y-axis is the absolute value of the x-coordinate, [tex]\(x\)[/tex].
- The distance of the point [tex]\((x, y)\)[/tex] from the origin [tex]\((0, 0)\)[/tex] can be found using the distance formula: [tex]\(\sqrt{x^2 + y^2}\)[/tex].
2. Given Condition:
- According to the problem, the distance from the Y-axis (i.e., [tex]\(|x|\)[/tex]) is half of its distance from the origin (i.e., [tex]\(\sqrt{x^2 + y^2}\)[/tex]).
- Mathematically, this condition can be written as:
[tex]\[
|x| = \frac{1}{2} \sqrt{x^2 + y^2}
\][/tex]
3. Removing the Absolute Value:
- We know that the absolute value of [tex]\(x\)[/tex] is always non-negative. Considering the definition of the absolute value, we remove the absolute value by squaring both sides of the equation to eliminate the absolute value:
[tex]\[
(\sqrt{x^2 + y^2})^2 = (2|x|)^2
\][/tex]
This simplifies to:
[tex]\[
x^2 + y^2 = 4x^2
\][/tex]
4. Simplifying the Equation:
- Simplify the equation by combining like terms:
[tex]\[
x^2 + y^2 = 4x^2
\][/tex]
[tex]\[
y^2 = 4x^2 - x^2
\][/tex]
[tex]\[
y^2 = 3x^2
\][/tex]
5. Final Equation:
- We rearrange the equation to make it clearer:
[tex]\[
y^2 = 3x^2
\][/tex]
- This equation represents a hyperbola.
Thus, the equation of the locus of the point that moves so that its distance from the Y-axis is half of its distance from the origin is [tex]\(y^2 = 3x^2\)[/tex].
