STATEMENT-1 : If the length of subtangent and subnormal at point (x, y) on y = f(x) are 9 and 4 then x is equal to + 6.
and
STATEMENT-2 : Product of subtangent and subnormal is square of the ordinate of the point.

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is true, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we need to analyze the given statements about the subtangent and subnormal at a point on the curve \( y = f(x) \).

### Step-by-Step Solution:

1. **Understanding Subtangent and Subnormal**:
The subtangent (TG) and subnormal (GN) at a point \( (x, y) \) on the curve can be expressed in terms of the slope of the tangent line at that point. If \( \theta \) is the angle of the tangent with the x-axis, then:
- Subtangent \( TG = y \cdot \cot(\theta) \)
- Subnormal \( GN = y \cdot \tan(\theta) \)

2. **Using the Given Lengths**:
According to the problem, we have:
- \( TG = 9 \)
- \( GN = 4 \)

From the definitions, we can write:
\[ y \cdot \cot(\theta) = 9 \quad \text{(1)} \]
\[ y \cdot \tan(\theta) = 4 \quad \text{(2)} \]

3. **Expressing Cotangent and Tangent**:
Recall that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Let \( m = \tan(\theta) \), then \( \cot(\theta) = \frac{1}{m} \). Substituting this into equation (1):
\[ y \cdot \frac{1}{m} = 9 \implies y = 9m \quad \text{(3)} \]

4. **Substituting into the Second Equation**:
Substitute equation (3) into equation (2):
\[ (9m) \cdot m = 4 \implies 9m^2 = 4 \implies m^2 = \frac{4}{9} \implies m = \frac{2}{3} \text{ or } m = -\frac{2}{3} \]

5. **Finding the Value of \( y \)**:
Substitute \( m \) back into equation (3):
\[ y = 9m = 9 \cdot \frac{2}{3} = 6 \quad \text{or} \quad y = 9 \cdot -\frac{2}{3} = -6 \]
Thus, \( y = 6 \) or \( y = -6 \).

6. **Finding the Value of \( x \)**:
Now, we need to check if \( x \) can be determined. The problem states that \( x \) is equal to \( +6 \). However, we have not derived \( x \) from the information given. The relationship between \( x \) and \( y \) is not established, so we cannot conclude that \( x = 6 \) based solely on the values of \( y \).

7. **Conclusion on Statements**:
- **Statement 1**: "If the length of subtangent and subnormal at point \( (x, y) \) on \( y = f(x) \) are 9 and 4 then \( x \) is equal to +6." This statement is **false** because we cannot definitively determine \( x \) from the given information.
- **Statement 2**: "Product of subtangent and subnormal is square of the ordinate of the point." We can verify this:
\[ TG \cdot GN = 9 \cdot 4 = 36 = y^2 \]
Hence, this statement is **true**.

### Final Answer:
- Statement 1 is **False**.
- Statement 2 is **True**.