Consider the following statements in respect of two chords XY and ZT of a circle intersecting at P.
I. PX.PY = PZ.PT II. PXZ and PTY are similar triangles. Which of the statements given above is/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem regarding the two chords XY and ZT of a circle intersecting at point P, we will analyze the given statements step by step.

### Step-by-Step Solution:

1. **Understanding the Chords and Intersection**:
- We have two chords, XY and ZT, intersecting at point P inside the circle.
- Let PX be the length from P to X, PY be the length from P to Y, PZ be the length from P to Z, and PT be the length from P to T.

2. **Analyzing the First Statement**:
- The first statement claims that \( PX \cdot PY = PZ \cdot PT \).
- According to the intersecting chords theorem, when two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal. Therefore, this statement is true.

3. **Analyzing the Second Statement**:
- The second statement claims that triangles \( \triangle PXZ \) and \( \triangle PTY \) are similar.
- For triangles to be similar, their corresponding angles must be equal.
- Since both triangles share angle \( \angle P \), we need to check if the other angles are equal.
- By the properties of circles, angles subtended by the same arc are equal. Thus, \( \angle PXZ \) and \( \angle PTY \) are equal as they subtend the same arcs (XZ and TY respectively).
- Therefore, \( \triangle PXZ \sim \triangle PTY \) is also true.

4. **Conclusion**:
- Both statements I and II are correct.
- Thus, the answer to the question is that both statements are true.

### Final Answer:
Both statements are correct.