Line `3x + 6y = k` intersect the curve `2x^2 + 3y^2 + 2xy=1` at points A and B. Thecircle on AB as diameter passes through origin,then sum of all possible values of 'k' is

Options

A. 3
B. 4
C. `-4`
D. `-3`

Explanation

To solve the problem, we need to find the sum of all possible values of \( k \) such that the line \( 3x + 6y = k \) intersects the curve \( 2x^2 + 3y^2 + 2xy = 1 \) at points A and B, and the circle with diameter AB passes through the origin.

### Step 1: Rewrite the line equation
We start with the line equation:
\[ 3x + 6y = k \]
We can express \( y \) in terms of \( x \):
\[ y = \frac{k - 3x}{6} \]

### Step 2: Substitute \( y \) into the curve equation
Next, we substitute \( y \) into the curve equation:
\[ 2x^2 + 3y^2 + 2xy = 1 \]
Substituting \( y = \frac{k - 3x}{6} \):
\[ 2x^2 + 3\left(\frac{k - 3x}{6}\right)^2 + 2x\left(\frac{k - 3x}{6}\right) = 1 \]
Expanding this:
\[ 2x^2 + 3\left(\frac{(k - 3x)^2}{36}\right) + \frac{2x(k - 3x)}{6} = 1 \]
\[ 2x^2 + \frac{3(k^2 - 6kx + 9x^2)}{36} + \frac{2(kx - 3x^2)}{6} = 1 \]
\[ 2x^2 + \frac{k^2}{12} - \frac{kx}{6} + \frac{3x^2}{12} + \frac{kx}{3} - x^2 = 1 \]

### Step 3: Combine like terms
Combining like terms gives:
\[ \left(2 + \frac{3}{12} - 1\right)x^2 + \left(-\frac{k}{6} + \frac{k}{3}\right)x + \frac{k^2}{12} - 1 = 0 \]
This simplifies to:
\[ \left(\frac{8}{12}\right)x^2 + \left(\frac{k}{6}\right)x + \left(\frac{k^2}{12} - 1\right) = 0 \]
\[ \frac{2}{3}x^2 + \frac{k}{6}x + \left(\frac{k^2}{12} - 1\right) = 0 \]

### Step 4: Condition for intersection
For the line to intersect the curve at two points, the discriminant of this quadratic equation must be non-negative:
\[ D = \left(\frac{k}{6}\right)^2 - 4\left(\frac{2}{3}\right)\left(\frac{k^2}{12} - 1\right) \geq 0 \]
Calculating the discriminant:
\[ D = \frac{k^2}{36} - \frac{8}{36}(k^2 - 12) \geq 0 \]
\[ D = \frac{k^2}{36} - \frac{8k^2}{36} + \frac{96}{36} \geq 0 \]
\[ D = \frac{-7k^2 + 96}{36} \geq 0 \]
\[ -7k^2 + 96 \geq 0 \]
\[ 7k^2 \leq 96 \]
\[ k^2 \leq \frac{96}{7} \]
\[ k^2 \leq 13.7142857 \]
Taking square roots:
\[ -\sqrt{\frac{96}{7}} \leq k \leq \sqrt{\frac{96}{7}} \]

### Step 5: Finding the sum of possible values of \( k \)
The possible values of \( k \) are symmetric about zero, thus the sum of all possible values of \( k \) is:
\[ 0 \]

### Final Answer
The sum of all possible values of \( k \) is:
\[ \boxed{0} \]