If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the locus of the foot of perpendicular from O on AB is:

Options

A. `(x^(2) + y^(2))(x+y) = R^(2)xy`
B. `(x^(2) + y^(2))^(3) = 4R^(2)x^(2)y^(2)`
C. `(x^(2) + y^(2))^(2) + 4Rx^(2)y^(2)`
D. `(x^(2) + y^(2)) = 4R^(2)x^(2) y^(2)`

Explanation

To solve the problem, we need to find the locus of the foot of the perpendicular from the origin \( O(0, 0) \) to the line segment \( AB \), where \( A \) and \( B \) are the points where a circle of radius \( R \) intersects the coordinate axes.

### Step-by-Step Solution:

1. **Define Points A and B**:
Let the circle intersect the x-axis at point \( A(a, 0) \) and the y-axis at point \( B(0, b) \).

2. **Circle Equation**:
The equation of the circle with center \( (h, k) \) and radius \( R \) can be written as:
\[ (x - h)^2 + (y - k)^2 = R^2 \]
Since the circle passes through the origin \( O(0, 0) \), we can substitute \( (0, 0) \) into the equation:
\[ h^2 + k^2 = R^2 \quad \text{(1)} \]

3. **Coordinates of A and B**:
The coordinates of points \( A \) and \( B \) can also be expressed in terms of \( h \) and \( k \):
- For point \( A \): The distance from the center to point \( A \) is \( R \), so:
\[ (a - h)^2 + k^2 = R^2 \quad \text{(2)} \]
- For point \( B \): The distance from the center to point \( B \) is \( R \), so:
\[ h^2 + (b - k)^2 = R^2 \quad \text{(3)} \]

4. **Finding the Foot of the Perpendicular**:
Let \( P(h, k) \) be the foot of the perpendicular from the origin \( O(0, 0) \) to line \( AB \). The coordinates of \( P \) can be expressed as:
\[ P = \left( \frac{a}{\sqrt{a^2 + b^2}}, \frac{b}{\sqrt{a^2 + b^2}} \right) \]

5. **Using the Distance Formula**:
From the distance formula, we have:
\[ OP^2 = h^2 + k^2 \]
Substituting from equation (1):
\[ OP^2 = R^2 \]

6. **Substituting A and B**:
We can express \( a \) and \( b \) in terms of \( h \) and \( k \) using the equations derived from the circle:
\[ a = \frac{R^2}{h} \quad \text{and} \quad b = \frac{R^2}{k} \]

7. **Locus Equation**:
Substitute \( a \) and \( b \) into the equation \( a^2 + b^2 = 4R^2 \):
\[ \left(\frac{R^2}{h}\right)^2 + \left(\frac{R^2}{k}\right)^2 = 4R^2 \]
Simplifying gives:
\[ \frac{R^4}{h^2} + \frac{R^4}{k^2} = 4R^2 \]
Dividing through by \( R^2 \):
\[ \frac{R^2}{h^2} + \frac{R^2}{k^2} = 4 \]

8. **Final Locus Equation**:
Rearranging gives the final locus equation:
\[ x^2 + y^2 = 4R^2 \]
where \( x = h \) and \( y = k \).