STATEMENT-1 : From point (4, 0) three different normals can be drawn to the parabola `y^(2) =4x.`
and
STATEMENT-2 : From any point, atmost three different normals can be drawn to a hyperbola.

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 both statements regarding the normals drawn from a point to a parabola and a hyperbola.

### Step 1: Analyze Statement 1
We need to determine if three different normals can be drawn from the point (4, 0) to the parabola given by the equation \( y^2 = 4x \).

1. **Identify the parameters of the parabola**:
The equation \( y^2 = 4x \) is in the standard form of a parabola \( y^2 = 4ax \), where \( a = 1 \).

2. **Determine the point from which normals are drawn**:
The point is \( (4, 0) \), so \( k = 4 \).

3. **Check the condition for drawing three normals**:
The condition for three normals to be drawn from a point \( (k, 0) \) to the parabola is that \( k > 2a \).
- Here, \( 2a = 2 \times 1 = 2 \).
- Now, check if \( k > 2a \):
\[ 4 > 2 \quad \text{(True)} \]
Since this condition is satisfied, we can conclude that three different normals can indeed be drawn from the point (4, 0) to the parabola.

### Conclusion for Statement 1:
**Statement 1 is true**.

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### Step 2: Analyze Statement 2
We need to determine the maximum number of normals that can be drawn from any point to a hyperbola.

1. **Consider the general form of a hyperbola**:
The standard form of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

2. **Equation of the normal to the hyperbola**:
The equation of the normal at a point on the hyperbola can be expressed as:
\[ a \cos \theta \cdot x + b \cot \theta \cdot y = a^2 + b^2 \]
where \( \theta \) is the angle corresponding to the point on the hyperbola.

3. **Determine the number of normals**:
If we are drawing normals from a point \( (h, k) \), we substitute these values into the normal equation:
\[ a \cos \theta \cdot h + b \cot \theta \cdot k = a^2 + b^2 \]

4. **Transform the equation**:
By substituting \( \cos \theta \) and \( \cot \theta \) in terms of a parameter \( t \) (where \( t = \tan(\theta/2) \)), we can derive a polynomial equation in \( t \).

5. **Degree of the polynomial**:
The resulting equation will be a polynomial of degree 4. A polynomial of degree 4 can have at most 4 real roots, which means that there can be at most 4 normals drawn from any point to the hyperbola.

### Conclusion for Statement 2:
**Statement 2 is false** because the maximum number of normals that can be drawn from any point to a hyperbola is 4.

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### Final Conclusion:
- **Statement 1 is true**.
- **Statement 2 is false**.

Thus, the correct answer is that Statement 1 is true and Statement 2 is false.

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