If the points A(2, 5) and B are symmetrical about the tangent to the circle `x^(2)+y^(2)-4x+4y=0` at the origin, then the coordinates of B, are

Options

A. (5, -2)
B. (1, 5)
C. (5, 2)
D. none of these

Answer

To solve the problem step by step, we will first find the equation of the tangent to the given circle at the origin and then determine the coordinates of point B which is symmetrical to point A about this tangent.

### Step 1: Write the equation of the circle
The given equation of the circle is:
\[ x^2 + y^2 - 4x + 4y = 0 \]

### Step 2: Rearranging the equation
We can rearrange the equation to identify the center and radius:
\[ x^2 - 4x + y^2 + 4y = 0 \]
Completing the square for \(x\) and \(y\):
\[ (x - 2)^2 - 4 + (y + 2)^2 - 4 = 0 \]
\[ (x - 2)^2 + (y + 2)^2 = 8 \]
This shows that the center of the circle is at (2, -2) and the radius is \( \sqrt{8} \).

### Step 3: Find the slope of the tangent at the origin
To find the slope of the tangent at the origin, we differentiate the circle's equation implicitly:
\[ 2x + 2y \frac{dy}{dx} - 4 + 4 \frac{dy}{dx} = 0 \]
At the origin (0, 0), substituting \(x = 0\) and \(y = 0\):
\[ -4 + 4 \frac{dy}{dx} = 0 \]
Thus,
\[ 4 \frac{dy}{dx} = 4 \]
\[ \frac{dy}{dx} = 1 \]
This means the slope of the tangent line at the origin is 1.

### Step 4: Write the equation of the tangent line
The equation of the tangent line at the origin with slope 1 is:
\[ y = x \]

### Step 5: Find the coordinates of point B
Point A is given as \(A(2, 5)\). To find point B, we need to find its coordinates such that it is symmetrical to A about the line \(y = x\).

The midpoint M of A and B must lie on the line \(y = x\). Let the coordinates of B be \(B(h, k)\). The midpoint M is given by:
\[ M\left(\frac{2 + h}{2}, \frac{5 + k}{2}\right) \]

For M to lie on the line \(y = x\), we have:
\[ \frac{5 + k}{2} = \frac{2 + h}{2} \]
This simplifies to:
\[ 5 + k = 2 + h \]
\[ k = h - 3 \quad \text{(1)} \]

### Step 6: Use the symmetry condition
Since A and B are symmetrical about the line \(y = x\), we can also use the property that the coordinates of A and B will swap when reflected across this line:
\[ h = 5 \quad \text{and} \quad k = 2 \]
Thus, substituting \(h = 5\) into equation (1):
\[ k = 5 - 3 = 2 \]

### Step 7: Conclusion
The coordinates of point B are:
\[ B(5, 2) \]

### Final Answer
The coordinates of point B are \( (5, 2) \).

---