A point on the line x=4 from which the tangents drawn to the circle `2(x^(2)+y^(2))=25` are at right angles, is

Options

A. (4, 3)
B. (4, 4)
C. (4, -4)
D. none of these

Explanation

To solve the problem step by step, we need to find the point on the line \( x = 4 \) from which the tangents drawn to the circle \( 2(x^2 + y^2) = 25 \) are at right angles.

### Step 1: Rewrite the equation of the circle
The given equation of the circle is:
\[ 2(x^2 + y^2) = 25 \]
Dividing both sides by 2, we get:
\[ x^2 + y^2 = \frac{25}{2} \]

### Step 2: Identify the radius of the circle
From the equation \( x^2 + y^2 = \frac{25}{2} \), we can identify the radius \( r \) of the circle:
\[ r^2 = \frac{25}{2} \implies r = \sqrt{\frac{25}{2}} = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \]

### Step 3: Write the equation of the director circle
The equation of the director circle is given by:
\[ x^2 + y^2 = 2r^2 \]
Substituting \( r^2 = \frac{25}{2} \):
\[ x^2 + y^2 = 2 \times \frac{25}{2} = 25 \]

### Step 4: Set up the point on the line \( x = 4 \)
Let the point from which the tangents are drawn be \( P(h, k) \). Since the point lies on the line \( x = 4 \), we have:
\[ h = 4 \]
Thus, the coordinates of point \( P \) can be written as \( (4, k) \).

### Step 5: Use the condition for tangents being perpendicular
For the tangents drawn from point \( P(4, k) \) to the circle to be at right angles, the following condition must hold:
\[ h^2 + k^2 = 25 \]
Substituting \( h = 4 \):
\[ 4^2 + k^2 = 25 \]
This simplifies to:
\[ 16 + k^2 = 25 \]
\[ k^2 = 25 - 16 = 9 \]
Taking the square root gives:
\[ k = \pm 3 \]

### Step 6: Write the coordinates of point \( P \)
Thus, the coordinates of point \( P \) are:
\[ P(4, 3) \quad \text{or} \quad P(4, -3) \]

### Conclusion
The points on the line \( x = 4 \) from which the tangents drawn to the circle are at right angles are \( (4, 3) \) and \( (4, -3) \).

### Final Answer
The required points are \( (4, 3) \) and \( (4, -3) \).

---