IF the equation of conic
`2x^2+xy+3y^2-3x+5y+lamda=0` represent a single point, then find the value of `lamda`

Answer

To find the value of \( \lambda \) such that the conic equation \( 2x^2 + xy + 3y^2 - 3x + 5y + \lambda = 0 \) represents a single point, we need to analyze the condition for the equation to represent a pair of coincident lines. This occurs when the discriminant of the conic is zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The general form of a conic is given by:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
For our equation:
\[ 2x^2 + xy + 3y^2 - 3x + 5y + \lambda = 0 \]
We can identify the coefficients as:
- \( A = 2 \)
- \( B = 1 \)
- \( C = 3 \)
- \( D = -3 \)
- \( E = 5 \)
- \( F = \lambda \)

2. **Set up the discriminant condition**:
The condition for the conic to represent a single point (or coincident lines) is given by the discriminant \( \Delta \) being equal to zero:
\[ \Delta = B^2 - 4AC = 0 \]

3. **Calculate the discriminant**:
Substitute the values of \( A \), \( B \), and \( C \):
\[ \Delta = 1^2 - 4 \cdot 2 \cdot 3 = 1 - 24 = -23 \]
Since this is not zero, we need to consider the complete discriminant condition that includes the terms involving \( D \), \( E \), and \( F \).

4. **Use the complete discriminant formula**:
The complete discriminant for a conic section is given by:
\[ \Delta = A \cdot B \cdot C + 2 \cdot D \cdot E \cdot F - A \cdot E^2 - B \cdot D^2 - C \cdot F^2 \]
Substituting the values:
\[ \Delta = 2 \cdot 1 \cdot 3 + 2 \cdot (-3) \cdot 5 \cdot \lambda - 2 \cdot 5^2 - 1 \cdot (-3)^2 - 3 \cdot \lambda^2 \]

5. **Simplify the expression**:
\[ \Delta = 6 - 30\lambda - 50 - 9 - 3\lambda^2 \]
\[ \Delta = -3\lambda^2 - 30\lambda - 53 \]

6. **Set the discriminant to zero**:
\[ -3\lambda^2 - 30\lambda - 53 = 0 \]
To simplify, multiply through by -1:
\[ 3\lambda^2 + 30\lambda + 53 = 0 \]

7. **Use the quadratic formula**:
The quadratic formula is given by:
\[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = 30 \), and \( c = 53 \):
\[ \lambda = \frac{-30 \pm \sqrt{30^2 - 4 \cdot 3 \cdot 53}}{2 \cdot 3} \]
\[ = \frac{-30 \pm \sqrt{900 - 636}}{6} \]
\[ = \frac{-30 \pm \sqrt{264}}{6} \]
\[ = \frac{-30 \pm 2\sqrt{66}}{6} \]
\[ = \frac{-15 \pm \sqrt{66}}{3} \]

8. **Final value of \( \lambda \)**:
The values of \( \lambda \) that satisfy the condition are:
\[ \lambda = \frac{-15 + \sqrt{66}}{3} \quad \text{or} \quad \lambda = \frac{-15 - \sqrt{66}}{3} \]