For What value of `lamda` the equation of conic 2xy+4x-6y+`lamda`=0 represents two intersecting straight lines, if `lamda=17`, then this equation represents?

Answer

To solve the problem step-by-step, we will analyze the given conic equation and find the required values of λ.

### Step 1: Understand the given conic equation
The equation provided is:
\[ 2xy + 4x - 6y + \lambda = 0 \]

### Step 2: Identify the general form of the conic
The general form of a conic section is:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]
We need to identify the coefficients \(a\), \(h\), \(b\), \(g\), \(f\), and \(c\) from the given equation.

### Step 3: Compare coefficients
From the equation \(2xy + 4x - 6y + \lambda = 0\), we can compare it with the general form:
- \(a = 0\) (coefficient of \(x^2\))
- \(h = 1\) (coefficient of \(xy\))
- \(b = 0\) (coefficient of \(y^2\))
- \(g = 2\) (coefficient of \(x\))
- \(f = -3\) (coefficient of \(y\))
- \(c = \lambda\)

### Step 4: Set up the determinant condition
For the conic to represent two intersecting straight lines, the determinant must be zero:
\[ D = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0 \]

Substituting the values we found:
\[ D = \begin{vmatrix} 0 & 1 & 2 \\ 1 & 0 & -3 \\ 2 & -3 & \lambda \end{vmatrix} \]

### Step 5: Calculate the determinant
Calculating the determinant:
\[ D = 0 \cdot \begin{vmatrix} 0 & -3 \\ -3 & \lambda \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & -3 \\ 2 & \lambda \end{vmatrix} + 2 \cdot \begin{vmatrix} 1 & 0 \\ 2 & -3 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \(\begin{vmatrix} 1 & -3 \\ 2 & \lambda \end{vmatrix} = 1 \cdot \lambda - (-3) \cdot 2 = \lambda + 6\)
2. \(\begin{vmatrix} 1 & 0 \\ 2 & -3 \end{vmatrix} = 1 \cdot (-3) - 0 \cdot 2 = -3\)

Putting it all together:
\[ D = 0 - (\lambda + 6) + 2(-3) = -\lambda - 6 - 6 = -\lambda - 12 \]

### Step 6: Set the determinant to zero
Setting the determinant to zero for the conic to represent two intersecting lines:
\[ -\lambda - 12 = 0 \]
Thus,
\[ \lambda = -12 \]

### Step 7: Analyze the case when \(\lambda = 17\)
Now, we need to check what the conic represents when \(\lambda = 17\):
Substituting \(\lambda = 17\) into the equation:
\[ 2xy + 4x - 6y + 17 = 0 \]

### Step 8: Check the conditions for conic sections
We will check the condition \(h^2 - ab\):
- \(h = 1\), \(a = 0\), \(b = 0\)
- \(h^2 - ab = 1^2 - (0)(0) = 1\)

Since \(h^2 - ab > 0\), the conic represents a hyperbola.

### Final Answer
Thus, the value of \(\lambda\) for which the equation represents two intersecting straight lines is \(-12\), and when \(\lambda = 17\), the equation represents a hyperbola.

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