A circle touches a given straight line and cuts off a constant length 2d from another straight line perpendicular to the first straight line. The locus of the centre of the circle, is

Options

A. `y^(2)-x^(2)=d^(2)`
B. `x^(2)+y^(2)=d^(2)`
C. `xy=d^(2)`
D. none of these

Explanation

To solve the problem, we need to find the locus of the center of a circle that touches a given straight line and cuts off a constant length from another straight line perpendicular to the first. Let's break this down step by step.

### Step 1: Understand the Configuration
Assume the first straight line is the x-axis (y = 0). The second straight line, which is perpendicular to the x-axis, can be represented as a vertical line (for example, the line x = k).

**Hint:** Visualize the setup with the x-axis and a vertical line.

### Step 2: Define the Circle
Let the center of the circle be at point O(g, f). The radius of the circle will be the distance from the center to the x-axis, which is |f|. Since the circle touches the x-axis, the radius is equal to the distance from the center to the x-axis.

**Hint:** Remember that the radius is the distance from the center to the point of tangency.

### Step 3: Circle Equation
The general equation of the circle can be written as:
\[ (x - g)^2 + (y - f)^2 = r^2 \]
Since the circle touches the x-axis, we have:
\[ r = |f| \]
Thus, the equation becomes:
\[ (x - g)^2 + (y - f)^2 = f^2 \]

**Hint:** Relate the radius of the circle to the distance from the center to the x-axis.

### Step 4: Length Cut Off by the Circle
The problem states that the circle cuts off a constant length of 2d from the line perpendicular to the x-axis. The distance from the center of the circle to the line x = k is |g - k|. The length intercepted by the circle on this line is given by:
\[ 2\sqrt{f^2 - (g - k)^2} = 2d \]

**Hint:** Use the formula for the length of a chord in a circle.

### Step 5: Set Up the Equation
From the equation above, we can simplify:
\[ \sqrt{f^2 - (g - k)^2} = d \]
Squaring both sides gives:
\[ f^2 - (g - k)^2 = d^2 \]
Rearranging gives:
\[ f^2 = d^2 + (g - k)^2 \]

**Hint:** This equation relates the variables f, g, and d.

### Step 6: Locus of the Center
To find the locus of the center (g, f), we need to eliminate k. Since k can vary, we can express the relationship between f and g:
\[ f^2 - g^2 = d^2 \]
This is a hyperbola in the form:
\[ f^2 - g^2 = d^2 \]

**Hint:** Recognize that this equation represents a hyperbola.

### Final Result
The locus of the center of the circle is given by the equation:
\[ y^2 - x^2 = d^2 \]
where we have replaced f with y and g with x.

### Conclusion
Thus, the locus of the center of the circle is:
\[ y^2 - x^2 = d^2 \]

**Hint:** The final equation represents the required locus of the center of the circle.