All chords.of the curve `x^2+y^2-10x-4y+4=0` which make a right angle at (8,-2) pass through

Options

A. `(2,5)`
B. `(-2,-5)`
C. `(-5,-2)`
D. `(5,2)`

Explanation

To solve the problem, we need to find all chords of the circle defined by the equation \( x^2 + y^2 - 10x - 4y + 4 = 0 \) that make a right angle at the point \( (8, -2) \).

### Step-by-Step Solution:

1. **Rewrite the Circle Equation**:
The given equation of the circle is:
\[ x^2 + y^2 - 10x - 4y + 4 = 0 \]
We will complete the square for both \( x \) and \( y \).

- For \( x \):
\[ x^2 - 10x = (x - 5)^2 - 25 \]
- For \( y \):
\[ y^2 - 4y = (y - 2)^2 - 4 \]

Substituting back into the equation, we get:
\[ (x - 5)^2 - 25 + (y - 2)^2 - 4 + 4 = 0 \]
Simplifying gives:
\[ (x - 5)^2 + (y - 2)^2 = 25 \]

This is the equation of a circle with center \( (5, 2) \) and radius \( 5 \).

2. **Verify the Point Lies on the Circle**:
We need to check if the point \( (8, -2) \) lies on the circle:
\[ (8 - 5)^2 + (-2 - 2)^2 = 3^2 + (-4)^2 = 9 + 16 = 25 \]
Since this is true, the point \( (8, -2) \) lies on the circle.

3. **Understanding the Chord Condition**:
The problem states that all chords of the circle that make a right angle at the point \( (8, -2) \) must pass through the center of the circle. This is because a chord that makes a right angle at a point on the circle is a diameter of the circle.

4. **Identify the Center of the Circle**:
From our earlier work, we found that the center of the circle is \( (5, 2) \).

5. **Conclusion**:
Therefore, all chords of the circle that make a right angle at the point \( (8, -2) \) pass through the center of the circle, which is \( (5, 2) \).

### Final Answer:
The point through which all such chords pass is \( (5, 2) \).