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Class 11 Maths Chapter 10: Conic Sections case study question with answers

One case-based question of the kind CBSE-pattern school exams ask: a passage, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.

Case study · 4 marks
A park in Mysuru is mapped on a coordinate plane with the main gate at the origin O and distances in metres. The edge of a circular fountain in the park is the curve x2 + y2 − 12x − 16y + 75 = 0. The gardener wants to know where the fountain sits, how far its rim is from the gate, and which of the park's fixtures stand inside the fountain's edge, so that the water does not reach the electric fittings.
Part 1 · 1 mark
The radius of the fountain, in metres, is
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Answer: (b) 5

(x − 6)2 + (y − 8)2 = 36 + 64 − 75 = 25: centre (6, 8), radius 5 m.

Part 2 · 1 mark
The distance from the gate O to the farthest point of the fountain's edge, in metres, is
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Answer: (d) 15

O is √(36 + 64) = 10 m from the centre, outside the circle; the farthest point of the edge is on the line through O and the centre, 10 + 5 = 15 m away.

Part 3 · 2 marks
A lamp post stands at L(9, 10) and a bench at B(1, 4). Decide, for each, whether it is inside, on or outside the fountain's edge.
Answer and marking scheme

For L: (9 − 6)2 + (10 − 8)2 = 9 + 4 = 13 < 25, so the lamp post is inside the edge. For B: (1 − 6)2 + (4 − 8)2 = 25 + 16 = 41 > 25, so the bench is outside.

  • L: 13 < 25, inside (1)
  • B: 41 > 25, outside (1)
Or
A circular footpath is to be laid with the same centre as the fountain and a radius 2 m more than the fountain's. Find the equation of the footpath's edge in the form x2 + y2 + 2gx + 2fy + c = 0.
Answer and marking scheme

Centre (6, 8), radius 5 + 2 = 7: (x − 6)2 + (y − 8)2 = 49, i.e. x2 + y2 − 12x − 16y + 51 = 0.

  • centre (6, 8) and radius 7 (1)
  • x^{2} + y^{2} − 12x − 16y + 51 = 0 (1)
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