If `kx^(2)+10xy+3y^(2)-15x-21y+18=0` represents a pair of straight lines, then k=

Options

A. 3
B. 4
C. `-3`
D. none of these

Answer

To determine the value of \( k \) such that the equation \( kx^2 + 10xy + 3y^2 - 15x - 21y + 18 = 0 \) represents a pair of straight lines, we can use the condition for a conic section to represent a pair of straight lines. The condition is given by:

\[ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \]

where \( a, b, c, f, g, h \) are the coefficients from the general form of the conic section \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \).

### Step 1: Identify the coefficients
From the given equation, we can identify the coefficients:
- \( a = k \)
- \( b = 3 \)
- \( c = 18 \)
- \( h = 5 \) (since \( 10xy = 2hxy \))
- \( g = -\frac{15}{2} \) (from \( -15x = 2gx \))
- \( f = -\frac{21}{2} \) (from \( -21y = 2fy \))

### Step 2: Substitute the coefficients into the condition
Now we substitute these values into the condition:

\[ k \cdot 3 \cdot 18 + 2 \cdot \left(-\frac{21}{2}\right) \cdot \left(-\frac{15}{2}\right) \cdot 5 - k \left(-\frac{21}{2}\right)^2 - 3 \left(-\frac{15}{2}\right)^2 - 18 \cdot 5^2 = 0 \]

### Step 3: Calculate each term
1. Calculate \( k \cdot 3 \cdot 18 \):
\[ 54k \]

2. Calculate \( 2 \cdot \left(-\frac{21}{2}\right) \cdot \left(-\frac{15}{2}\right) \cdot 5 \):
\[ 2 \cdot \left(-\frac{21}{2}\right) \cdot \left(-\frac{15}{2}\right) \cdot 5 = \frac{21 \cdot 15 \cdot 5}{2} = \frac{1575}{2} \]

3. Calculate \( -k \left(-\frac{21}{2}\right)^2 \):
\[ -k \cdot \frac{441}{4} \]

4. Calculate \( -3 \left(-\frac{15}{2}\right)^2 \):
\[ -3 \cdot \frac{225}{4} = -\frac{675}{4} \]

5. Calculate \( -18 \cdot 5^2 \):
\[ -18 \cdot 25 = -450 \]

### Step 4: Combine all terms
Putting it all together, we have:
\[ 54k + \frac{1575}{2} - k \cdot \frac{441}{4} - \frac{675}{4} - 450 = 0 \]

### Step 5: Clear the fractions
Multiply through by 4 to eliminate the fractions:
\[ 216k + 3150 - 441k - 675 - 1800 = 0 \]

### Step 6: Simplify the equation
Combine like terms:
\[ (216k - 441k) + (3150 - 675 - 1800) = 0 \]
\[ -225k + 675 = 0 \]

### Step 7: Solve for \( k \)
\[ -225k = -675 \]
\[ k = \frac{675}{225} = 3 \]

### Conclusion
Thus, the value of \( k \) is \( 3 \).