If the ratio of gradients of the line given by `ax^(2)+2hxy+by^(2)=0` is `1:3`, then `h^(2):ab=`

Options

A. `(1)/(3)`
B. `1`
C. `(3)/(4)`
D. `(4)/(3)`

Answer

To solve the problem, we need to find the ratio \( h^2 : ab \) given the equation of a pair of straight lines \( ax^2 + 2hxy + by^2 = 0 \) and the ratio of the gradients (slopes) of the lines is \( 1:3 \).

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines through the origin.

2. **Substituting for Slope**:
We can express \( y \) in terms of \( x \) by letting \( y = mx \) (where \( m \) is the slope). Substituting this into the equation gives:
\[ a(x^2) + 2h(mx)x + b(mx)^2 = 0 \]
Simplifying this results in:
\[ ax^2 + 2hmx^2 + bmx^2 = 0 \]
Factoring out \( x^2 \) (assuming \( x \neq 0 \)):
\[ (a + 2hm + bm^2) = 0 \]

3. **Forming the Quadratic Equation**:
This leads to the quadratic equation in terms of \( m \):
\[ bm^2 + 2hm + a = 0 \]

4. **Finding the Roots**:
The roots \( m_1 \) and \( m_2 \) of this quadratic equation can be found using the quadratic formula:
\[ m = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Here, \( A = b \), \( B = 2h \), and \( C = a \). Thus, the slopes are:
\[ m_1, m_2 = \frac{-2h \pm \sqrt{(2h)^2 - 4ab}}{2b} \]

5. **Using the Given Ratio**:
The problem states that the ratio of the slopes is \( m_1 : m_2 = 1 : 3 \). This implies:
\[ m_2 = 3m_1 \]
From the sum of the roots \( m_1 + m_2 = -\frac{2h}{b} \):
\[ m_1 + 3m_1 = -\frac{2h}{b} \implies 4m_1 = -\frac{2h}{b} \implies m_1 = -\frac{2h}{4b} = -\frac{h}{2b} \]

6. **Finding the Product of the Roots**:
The product of the roots \( m_1 m_2 = \frac{a}{b} \):
\[ m_1 \cdot 3m_1 = \frac{a}{b} \implies 3m_1^2 = \frac{a}{b} \]
Substituting \( m_1 = -\frac{h}{2b} \):
\[ 3\left(-\frac{h}{2b}\right)^2 = \frac{a}{b} \]
Simplifying gives:
\[ 3\frac{h^2}{4b^2} = \frac{a}{b} \]
Multiplying both sides by \( 4b^2 \):
\[ 3h^2 = 4ab \]

7. **Finding the Ratio \( h^2 : ab \)**:
Rearranging gives:
\[ \frac{h^2}{ab} = \frac{4}{3} \]
Therefore, the ratio \( h^2 : ab = 4 : 3 \).

### Final Answer:
\[ h^2 : ab = 4 : 3 \]