If the slope of one of the lines given by `36x^(2)+2hxy+72y^(2)=0` is four times the other, then `h^(2)=`

Options

A. 5040
B. 4050
C. 8100
D. none of these

Answer

To solve the problem, we need to find the value of \( h^2 \) given that the slopes of the lines represented by the equation \( 36x^2 + 2hxy + 72y^2 = 0 \) have a specific relationship (one slope is four times the other).

### Step-by-Step Solution:

1. **Identify the coefficients from the equation**:
The given equation is \( 36x^2 + 2hxy + 72y^2 = 0 \).
Here, we can identify:
- \( a = 36 \)
- \( b = 72 \)
- \( c = 2h \)

2. **Use the relationship between slopes**:
Let the slopes of the lines be \( m \) and \( k \). According to the problem, we have:
\[ m = 4k \]

3. **Sum of slopes**:
The sum of the slopes can be expressed using the formula:
\[ m + k = -\frac{b}{a} = -\frac{72}{36} = -2 \]
Substituting \( m = 4k \) into the equation gives:
\[ 4k + k = -2 \implies 5k = -2 \implies k = -\frac{2}{5} \]

4. **Find \( m \)**:
Now substituting back to find \( m \):
\[ m = 4k = 4 \left(-\frac{2}{5}\right) = -\frac{8}{5} \]

5. **Product of slopes**:
The product of the slopes is given by:
\[ mk = \frac{c}{a} = \frac{2h}{36} = \frac{h}{18} \]
Now substituting the values of \( m \) and \( k \):
\[ mk = \left(-\frac{8}{5}\right)\left(-\frac{2}{5}\right) = \frac{16}{25} \]

6. **Setting up the equation**:
Now equate the two expressions for \( mk \):
\[ \frac{h}{18} = \frac{16}{25} \]

7. **Solve for \( h \)**:
Cross-multiplying gives:
\[ 25h = 16 \times 18 \]
\[ 25h = 288 \implies h = \frac{288}{25} \]

8. **Find \( h^2 \)**:
Now, squaring \( h \):
\[ h^2 = \left(\frac{288}{25}\right)^2 = \frac{82944}{625} \]

9. **Final answer**:
Thus, the value of \( h^2 \) is:
\[ h^2 = \frac{82944}{625} \]