Statement I: The lines from the vertex to the two extremities of a focal chord of the parabola `y^2=4ax` are perpendicular to each other.
Statement II: If the extremities of focal chord of a parabola are `(at_1^2,2at_1)` and `(at_2^2,2at_2)`, then `t_1t_2=-1`.

Options

A. (a) Statement I is true, Statement II is true and Statement II is the correct explanation for Statement I.
B. (b) Statement I is true and Statement II is true but Statement II is not the correct explanation for Statement I.
C. (c) Statement I is true, Statement II is false.
D. (d) Statement I is false, Statement II is true.

Answer

To solve the given problem, we will analyze both statements regarding the parabola \( y^2 = 4ax \).

### Step 1: Understanding the Focal Chord
A focal chord of the parabola \( y^2 = 4ax \) has its endpoints at the points \( (at_1^2, 2at_1) \) and \( (at_2^2, 2at_2) \).

### Step 2: Verifying Statement II
According to Statement II, if the extremities of a focal chord are \( (at_1^2, 2at_1) \) and \( (at_2^2, 2at_2) \), then \( t_1 t_2 = -1 \).

To prove this, we can use the property of focal chords in a parabola. The product of the slopes of the lines connecting the vertex to the endpoints of the focal chord is given by:

\[ \text{slope of OA} = \frac{2at_1 - 0}{at_1^2 - 0} = \frac{2t_1}{t_1^2} = \frac{2}{t_1} \]
\[ \text{slope of OB} = \frac{2at_2 - 0}{at_2^2 - 0} = \frac{2t_2}{t_2^2} = \frac{2}{t_2} \]

The product of the slopes \( m_A \) and \( m_B \) is:

\[ m_A \cdot m_B = \left(\frac{2}{t_1}\right) \cdot \left(\frac{2}{t_2}\right) = \frac{4}{t_1 t_2} \]

For the lines to be perpendicular, this product must equal \(-1\):

\[ \frac{4}{t_1 t_2} = -1 \implies t_1 t_2 = -4 \]

This confirms that \( t_1 t_2 = -1 \) is indeed a property of the focal chord.

### Step 3: Analyzing Statement I
Now, we analyze Statement I which claims that the lines from the vertex to the two extremities of a focal chord are perpendicular to each other.

Using the slopes calculated earlier, we find:

\[ m_A = \frac{2}{t_1}, \quad m_B = \frac{2}{t_2} \]

For the lines to be perpendicular, the product of their slopes must equal \(-1\):

\[ m_A \cdot m_B = \left(\frac{2}{t_1}\right) \cdot \left(\frac{2}{t_2}\right) = \frac{4}{t_1 t_2} \]

Since we established that \( t_1 t_2 = -1 \):

\[ m_A \cdot m_B = \frac{4}{-1} = -4 \]

This indicates that the lines are not perpendicular, contradicting Statement I.

### Conclusion
- Statement I is **false**.
- Statement II is **true**.

Thus, the correct answer is **D: Statement I is false and Statement II is true**.

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