IPMAT Indore 2024 · Quantitative Ability · Question 29

The question does not say whether the point lies inside or outside the circle, so both cases must be considered. Let the radius be r and the distance from the point to the centre be d.

Case 1 — the point is outside. The nearest point of the circle is toward the centre, the farthest is beyond it:

d-r=4, d+r=9

Subtracting: 2r=5\Rightarrow r=2.5, and d=6.5. Check d>r ✔ (consistent: the point really is outside).

Case 2 — the point is inside. Now the shortest and longest distances are measured outward in opposite directions:

r-d=4, r+d=9

Adding: 2r=13\Rightarrow r=6.5, and d=2.5. Check d<r ✔ (consistent: the point really is inside).

Both cases are self-consistent, so both radii are possible.

Answer: Option 1 — 2.5 cm or 6.5 cm

Shortcut. Add and subtract — you get both answers at once.

\text{Sum}=9+4=13, \text{Difference}=9-4=5

Then r is half of one of them: 13/2=6.5 (point inside) or 5/2=2.5 (point outside), and d is half of the other.

The exam skill here is noticing the ambiguity, not the algebra. Any option offering a single value (b) or (d) is incomplete, and (c) just forgets to halve. When a geometry question withholds “inside” or “outside”, expect a two-answer option.