The locus of a point which moves so that its distance from x-axis is double of its distance from y-axis it

Options

A. `x=2y`
B. `y=2x`
C. `y=2x+3`
D. `x=5y+1`

Answer

To find the locus of a point that moves such that its distance from the x-axis is double its distance from the y-axis, we can follow these steps:

### Step 1: Define the point
Let the point be \( P(h, k) \), where \( h \) is the x-coordinate and \( k \) is the y-coordinate.

### Step 2: Determine distances
- The distance from the point \( P \) to the x-axis is given by the absolute value of the y-coordinate, which is \( |k| \).
- The distance from the point \( P \) to the y-axis is given by the absolute value of the x-coordinate, which is \( |h| \).

### Step 3: Set up the relationship
According to the problem, the distance from the x-axis is double the distance from the y-axis. Therefore, we can write the equation:
\[ |k| = 2|h| \]

### Step 4: Remove absolute values
Since we are looking for a locus, we can consider both positive and negative cases:
1. \( k = 2h \)
2. \( k = -2h \)

### Step 5: Write the equations of the locus
From the equations derived in Step 4, we can express the relationship in terms of \( y \) and \( x \):
1. For \( k = 2h \):
\[ y = 2x \]
2. For \( k = -2h \):
\[ y = -2x \]

### Step 6: Combine the results
The locus of the point \( P \) is thus represented by the two lines:
\[ y = 2x \quad \text{and} \quad y = -2x \]

### Final Result
The locus of the point is the set of points that lie on the lines \( y = 2x \) and \( y = -2x \).

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