Given information:
- Two chords of lengths a and b subtend angles of 60° and 90° at the center of the circle, respectively.

To determine which of the given options is correct, let's analyze the given information and apply some properties of chords in a circle.

Properties of Chords in a Circle:
1. The perpendicular bisector of a chord passes through the center of the circle.
2. The angle subtended by a chord at the center of the circle is twice the angle subtended by the same chord at any point on the circumference.

Analysis:
1. The chord of length a subtends an angle of 60° at the center. Therefore, the angle subtended by this chord at any point on the circumference is 60°/2 = 30°.
2. The chord of length b subtends an angle of 90° at the center. Therefore, the angle subtended by this chord at any point on the circumference is 90°/2 = 45°.

Now, let's analyze the given options one by one:

Option A: b = 2a
- According to the properties mentioned above, the angle subtended by the chord of length b at any point on the circumference is 45°.
- If b = 2a, then the angle subtended by the chord of length b at any point on the circumference should be 2 times the angle subtended by the chord of length a.
- However, 45° is not 2 times 30°.
- Therefore, Option A is incorrect.

Option B: b = 2b
- This option is incorrect because it is comparing the length of a chord with itself.

Option C: a = 2b
- According to the properties mentioned above, the angle subtended by the chord of length a at any point on the circumference is 30°.
- If a = 2b, then the angle subtended by the chord of length a at any point on the circumference should be 2 times the angle subtended by the chord of length b.
- However, 30° is not 2 times 45°.
- Therefore, Option C is incorrect.

Option D: b = 2a
- According to the properties mentioned above, the angle subtended by the chord of length b at any point on the circumference is 45°.
- If b = 2a, then the angle subtended by the chord of length b at any point on the circumference should be 2 times the angle subtended by the chord of length a.
- 45° is indeed 2 times 30°.
- Therefore, Option D is correct.

Therefore, the correct answer is Option D: b = 2a.