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XY and X′Y′ are parallel… (Class 10 Maths 2025, Q31)

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MathsCBSE Class 10 · Standard · Board exam 2025, set 30/5/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 31

Set in the CBSE Class 10 board exam of 2025, paper 30/5/1, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerCircles

In the adjoining figure, XY and X′Y′ are parallel tangents to a circle with centre O. Another tangent AB touches the circle at C intersecting XY at A and X′Y′ at B. Prove that AB subtends right angle at the centre of the circle; or .

Figure in the paper: A circle with centre O. Two horizontal parallel tangent lines XY (top, touching at P) and X′Y′ (bottom, touching at Q), with PQ a vertical diameter through O. Points A on the top tangent and B on the bottom tangent; the slanted tangent AB touches the circle at C between A and B. Radii OA and OB are drawn and angle AOB is marked at O. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/5/1.

Hints

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

Hint 1

Join OC and look at the pairs of triangles OPA, OCA and OQB, OCB.

Hint 2

Use: tangents from an external point are equal, and radii are equal.

Hint 3

The angles at O on one side of diameter PQ add up to 180°.

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.

  1. Join OC, OP and OQ. Since and [tangent ⟂ radius] and , the points P, O, Q are collinear, so POQ is a diameter.

    ½ mark

  2. In and : [radii], [tangents from an external point are equal] and OA is common. So [SSS].

    ½ mark

  3. [CPCT] ...(1)

    ½ mark

  4. Similarly, , so ...(2)

    ½ mark

  5. POQ is a straight line, so .

    ½ mark

  6. Using (1) and (2): . Hence .

    ½ mark

Answer: (proved)

Where the marks are

  • Joins OC (with PQ as diameter) (½ mark)
  • Proves with reasons (½ mark)
  • Gets (½ mark)
  • Similarly gets (½ mark)
  • Sum of the four angles at O on line PQ = 180° (½ mark)
  • Concludes (½ mark)

Revise the chapter

Practise this chapterBoard-style questions from Circles, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 30/5/1 (Standard): 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.