If a chord of a circle `x ^(2) + y ^(2) = 25` with one extermity at (4,3) substends a right angle at the centre of this circle, then the coordinates of te other extermity of this chord can be.

Options

A. `(-4,-4)`
B. `(4,-3)`
C. `(3,4)`
D. `(-3,4)`

Answer

To find the coordinates of the other extremity of the chord that subtends a right angle at the center of the circle defined by the equation \(x^2 + y^2 = 25\), we can follow these steps:

### Step 1: Identify the center and radius of the circle
The given equation of the circle is \(x^2 + y^2 = 25\). This represents a circle centered at the origin \((0, 0)\) with a radius of \(5\) (since \(\sqrt{25} = 5\)).

### Step 2: Define the points
Let \(A(4, 3)\) be one extremity of the chord, and let \(B(x_1, y_1)\) be the other extremity of the chord. The center of the circle is \(O(0, 0)\).

### Step 3: Use the property of perpendicular chords
Since the chord \(AB\) subtends a right angle at the center \(O\), the slopes of \(OA\) and \(OB\) must satisfy the condition:
\[ \text{slope of } OA \times \text{slope of } OB = -1 \]

The slope of line \(OA\) is given by:
\[ \text{slope of } OA = \frac{y_A - y_O}{x_A - x_O} = \frac{3 - 0}{4 - 0} = \frac{3}{4} \]

Let the slope of line \(OB\) be \(\frac{y_1 - 0}{x_1 - 0} = \frac{y_1}{x_1}\). Therefore, we have:
\[ \frac{3}{4} \cdot \frac{y_1}{x_1} = -1 \]

### Step 4: Solve for \(y_1\)
From the equation above, we can rearrange to find \(y_1\):
\[ y_1 = -\frac{4}{3} x_1 \]

### Step 5: Use the circle equation for point \(B\)
Since point \(B(x_1, y_1)\) lies on the circle, we can substitute \(y_1\) into the circle's equation:
\[ x_1^2 + y_1^2 = 25 \]
Substituting \(y_1 = -\frac{4}{3} x_1\):
\[ x_1^2 + \left(-\frac{4}{3} x_1\right)^2 = 25 \]
\[ x_1^2 + \frac{16}{9} x_1^2 = 25 \]
\[ \left(1 + \frac{16}{9}\right)x_1^2 = 25 \]
\[ \frac{25}{9} x_1^2 = 25 \]
Multiplying both sides by \(\frac{9}{25}\):
\[ x_1^2 = 9 \]
Taking the square root:
\[ x_1 = 3 \quad \text{or} \quad x_1 = -3 \]

### Step 6: Find corresponding \(y_1\) values
1. If \(x_1 = 3\):
\[ y_1 = -\frac{4}{3} \cdot 3 = -4 \]
Thus, one point is \(B(3, -4)\).

2. If \(x_1 = -3\):
\[ y_1 = -\frac{4}{3} \cdot (-3) = 4 \]
Thus, the other point is \(B(-3, 4)\).

### Conclusion
The coordinates of the other extremity of the chord can be either \((3, -4)\) or \((-3, 4)\).