If the common chord of the circles `x^(2) + ( y -lambda)^(2) =16` and `x^(2) +y^(2) =16` subtend a right angle at the origin then `' lambda'` is equal to `:`

Options

A. 4
B. `2 sqrt( 2)`
C. `+- 4sqrt( 2)`
D. 8

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the common chord of the circles \( x^2 + (y - \lambda)^2 = 16 \) and \( x^2 + y^2 = 16 \) subtends a right angle at the origin.

### Step 1: Write down the equations of the circles.
The equations of the circles are:
1. Circle 1: \( x^2 + (y - \lambda)^2 = 16 \)
2. Circle 2: \( x^2 + y^2 = 16 \)

### Step 2: Find the equation of the common chord.
The equation of the common chord can be found using the formula \( S - S' = 0 \), where \( S \) is the equation of the first circle and \( S' \) is the equation of the second circle.

Substituting the equations:
- \( S = x^2 + (y - \lambda)^2 - 16 = 0 \)
- \( S' = x^2 + y^2 - 16 = 0 \)

Thus, we have:
\[ S - S' = (x^2 + (y - \lambda)^2 - 16) - (x^2 + y^2 - 16) = 0 \]
This simplifies to:
\[ (y - \lambda)^2 - y^2 = 0 \]
Expanding \( (y - \lambda)^2 \):
\[ y^2 - 2\lambda y + \lambda^2 - y^2 = 0 \]
This simplifies to:
\[ -2\lambda y + \lambda^2 = 0 \]
Rearranging gives:
\[ 2\lambda y = \lambda^2 \]
So, we can express \( y \) as:
\[ y = \frac{\lambda}{2} \]

### Step 3: Condition for the chord to subtend a right angle at the origin.
For the common chord to subtend a right angle at the origin, the following condition must hold:
\[ x^2 + y^2 - 16 = 0 \]
This means that the line represented by the common chord must satisfy the condition that the coefficients of \( x^2 \) and \( y^2 \) must sum to zero.

### Step 4: Substitute \( y \) into the equation.
Substituting \( y = \frac{\lambda}{2} \) into the equation \( x^2 + y^2 - 16 = 0 \):
\[ x^2 + \left(\frac{\lambda}{2}\right)^2 - 16 = 0 \]
This gives:
\[ x^2 + \frac{\lambda^2}{4} - 16 = 0 \]
Rearranging gives:
\[ x^2 = 16 - \frac{\lambda^2}{4} \]

### Step 5: Set up the right angle condition.
For the chord to subtend a right angle at the origin, we need:
\[ \frac{2}{\lambda} = 1 \]
This leads to:
\[ \lambda^2 = 32 \]

### Step 6: Solve for \( \lambda \).
Taking the square root gives:
\[ \lambda = \pm 4\sqrt{2} \]

### Final Answer:
Thus, the value of \( \lambda \) is:
\[ \lambda = 4\sqrt{2} \text{ or } \lambda = -4\sqrt{2} \]