MathsCBSE Class 10 · Basic · Board exam 2026, set 430/1/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2026: question 38

Set in the CBSE Class 10 board exam of 2026, paper 430/1/1, section E, question 38, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q38 · 4 marks · Case-basedCompetency-basedCoordinate Geometry

Case Study – 3

In a society, there is a circular park having two gates. The gates are placed at points A(10, 20) and B(50, 50), as shown in the figure below. Two fountains are installed at points P and Q on AB such that $A P = P Q = QB$ .

Based on the above information, answer the following questions :

(i)
Find the coordinates of the centre C.
1 mark
(ii)
Find the radius of the circular park.
1 mark
(iii)
(a) Find the coordinates of the point P.
2 marks
OR
(iii)
(b) Find the distance of the fountain at Q from gate A.
2 marks

Figure in the paper: Circle representing the park with a diameter AB; A is at lower left on the circle and B at upper right on the circle. C is the centre on AB. P and Q lie on AB between A and C and between C and B respectively, with AP = PQ = QB. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 430/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

AB is a diameter, so the centre is its midpoint.

Hint 2

The radius is half the length of AB; use the distance formula.

Hint 3

AP = PQ = QB means P and Q trisect AB: P divides it in the ratio 1:2 and Q in the ratio 2:1.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) AB is a diameter, so C is its midpoint: $C = (\frac{10 + 50}{2}, \frac{20 + 50}{2}) = (30, 35)$ .
1 mark
(ii) $A B = (50 - 10)^{2} + (50 - 20)^{2} = 1600 + 900 = 50$ units, so radius $= \frac{50}{2} = 25$ units.
1 mark
(iii)(a) Since $A P = P Q = QB$ , P divides AB in the ratio $1 : 2$ . By the section formula, $x = \frac{1 ( 50 ) + 2 ( 10 )}{3}$ and $y = \frac{1 ( 50 ) + 2 ( 20 )}{3}$ .
1 mark
(iii)(a) $P = (\frac{70}{3}, 30) \approx (23.33, 30)$ .
1 mark
(iii)(b) Q divides AB in the ratio $2 : 1$ , so $Q = (\frac{2 ( 50 ) + 10}{3}, \frac{2 ( 50 ) + 20}{3}) = (\frac{110}{3}, 40)$ .
1 mark
(iii)(b) $A Q = (\frac{110}{3} - 10)^{2} + (40 - 20)^{2} = \frac{6400}{9} + \frac{3600}{9} = \frac{100}{3} \approx 33.33$ units. (Check: $A Q = \frac{2}{3} A B = \frac{2}{3} \times 50$ .)
1 mark

Answer: (i) C(30, 35); (ii) 25 units; (iii)(a) P(70/3, 30); (iii)(b) AQ = 100/3 ≈ 33.33 units

Where the marks are

(i) Midpoint formula gives C(30, 35) (1 mark)
(ii) AB = 50 by distance formula, so radius = 25 units (1 mark)
(iii)(a) Section formula with ratio 1:2 (1 mark)
(iii)(a) $P = (70/3, 30)$ (1 mark)
(iii)(b) Finds $Q = (110/3, 40)$ (1 mark)
(iii)(b) $A Q = 100/3$ units (1 mark)

Revise the chapter

Chapter: Coordinate Geometry (notes, formulas, where marks are lost and every past question)
Topic: Applications of the Distance Formula

Practise this chapterBoard-style questions from Coordinate Geometry, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 430/1/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2026: question 38

Hints

Step-by-step solution

Revise the chapter

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