The locus of a point which moves difference of its distance from two fixed straight which are at right angles is equal to the distance from another fixed straight line is

Options

A. a stright line
B. a circle
C. a parabola
D. an ellipse

Explanation

To solve the problem, we need to find the locus of a point that moves such that the difference of its distances from two fixed straight lines (which are at right angles) is equal to its distance from another fixed straight line.

Let's break down the solution step by step:

### Step 1: Define the Fixed Lines
Let the two fixed straight lines be:
1. Line 1: \( y = 0 \) (the x-axis)
2. Line 2: \( x = 0 \) (the y-axis)

These two lines are at right angles to each other.

### Step 2: Define the Variable Point
Let the variable point be \( P(h, k) \).

### Step 3: Calculate Distances
The distance from point \( P(h, k) \) to the x-axis (Line 1) is:
\[ d_1 = |k| \]
The distance from point \( P(h, k) \) to the y-axis (Line 2) is:
\[ d_2 = |h| \]

### Step 4: Set Up the Equation
According to the problem, the difference of the distances from the two lines is equal to the distance from another fixed line. Let’s assume the other fixed line is \( ax + by + c = 0 \). The distance from point \( P(h, k) \) to this line is given by:
\[ d = \frac{|ah + bk + c|}{\sqrt{a^2 + b^2}} \]

The condition given in the problem can be expressed as:
\[ |k| - |h| = \frac{|ah + bk + c|}{\sqrt{a^2 + b^2}} \]

### Step 5: Simplify the Equation
Assuming \( k \geq 0 \) and \( h \geq 0 \) (we can analyze other cases later), we can rewrite the equation as:
\[ k - h = \frac{ah + bk + c}{\sqrt{a^2 + b^2}} \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ k - \frac{bk}{\sqrt{a^2 + b^2}} = h + \frac{ah}{\sqrt{a^2 + b^2}} + \frac{c}{\sqrt{a^2 + b^2}} \]
\[ k \left(1 - \frac{b}{\sqrt{a^2 + b^2}}\right) = h \left(1 + \frac{a}{\sqrt{a^2 + b^2}}\right) + \frac{c}{\sqrt{a^2 + b^2}} \]

### Step 7: Identify the Locus
This equation represents a linear relationship between \( h \) and \( k \), indicating that the locus of the point is a straight line.

### Conclusion
The locus of the point that satisfies the given condition is a straight line.