UPSC CDS I 2025 Mathematics Paper

Given:
Circle M radius =8cm (centre U).
Circle N radius =16cm (centre V).
They touch externally ⇒ UV =8+16=24cm.
PQ= length of common external tangent =16√‌2cm.
Formula used:
If two radii to tangent points are perpendicular to the tangent, those two radii are parallel.
Area of a trapezoid with parallel sides a and b separated by distance d=(a+b)∕2×d.
Calculations:
⇒UP= radius of M=8cm(UP⟂PQ).
⇒VQ= radius of N=16cm(VQ⟂PQ).
⇒UP∕∕VQ and the distance between these parallel segments = length PQ=16√‌2cm.
⇒ Area of quadrilateral P−Q−V−U=(UP+VQ)∕2×PQ
⇒ Area =(8+16)∕2×16√‌2
⇒ Area =24∕2×16√‌2=12×16√‌2
⇒ Area =192√‌2cm2