I think it's pretty well explained from the Data Sufficiency angle. So, not solving the entire problem. But just adding a bit on the first part of the explanation related to the concepts of Intersecting Chords & Similarity. To understand the problem in a better way let us remember the following rules/ theorems.
1) Theorem related to Intersection of Chords: When two chords intersect each other inside a circle, then each is divided into 2 segments and the products of their segments are equal.
Thus in this question, when we consider the chords AB and PQ (note: Diameter is the largest chord of a circle), they intersect at X. Hence AX . BX = PX . QX
You can straightaway apply this to proceed with this question.
2) Theorem related to Angles subtended by an Arc on the Circle: Angles subtended on the circumference in the same segment of a circle by the same arc are always equal (note: These are call Internal Angles, and are half of the Central Angle subtended by the same arc at the centre of the circle).
Thus in this question if we consider the arc PA, it subtends two angles - one at B and another at Q.
Hence, angle(PAB) = angle(PQB)
Now if you consider the two triangles, PAX and QBX, then they have
angle(PAB) = angle(PQB) ..... as shown above
angle(PXA) = angle(QXB) ..... vertically opposite angles
And hence they are SIMILAR (note: two triangles are similar if all the three angles are equal and in fact if two sets of angles are equal, then automatically the third set of angles must also be equal).
Thus their "Corresponding Sides" will be in the same Ratio (note: by Corresponding sides I refer to the sides opposite to the equal angles in two similar triangles).
So, AX/ QX = PX/ BX
i.e. AX . BX = PX . QX
Again we arrive at the same results as in Point 1 and can proceed from there.
Although I would prefer a student to straightaway go for method 1, I thought of discussing the method 2 since the concept of Similarity was already discussed in context to this problem and I think all can develop good insights by understanding this approach.
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