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PA and PB are tangents… (Class 10 Maths SQP 2023-24, Q29a)

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MathsCBSE Class 10 · Standard · Sample paper, session 2023-24 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2023-24: question 29(a)

From the official CBSE sample question paper for 2023-24, section C, question 29(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(a) · 3 marks · Short answerCircles

PA and PB are tangents drawn to a circle of centre O from an external point P. Chord AB makes an angle of with the radius at the point of contact. If length of the chord is 6 cm, find the length of the tangent PA and the length of the radius OA.

Figure in the paper: Circle with centre O. External point P with tangents PA (touching at A, top) and PB (touching at B, bottom). Chord AB is drawn, and radii OA and OB are joined. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

The radius OA is perpendicular to the tangent PA. Use this to find ∠PAB.

Hint 2

Show that triangle PAB is equilateral. Then use triangle OAP, where ∠OPA = 30°.

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. Given . Also [tangent ⟂ radius at the point of contact].

    So .

    ½ mark

  2. [tangents from an external point are equal]. So [angles opposite equal sides].

    ½ mark

  3. In : [angle sum property]. So is equilateral.

    ½ mark

  4. Therefore cm.

    ½ mark

  5. OP bisects [centre lies on the bisector of the angle between the tangents], so .

    In right :

    ½ mark

  6. cm

    ½ mark

Answer: PA = 6 cm, OA = 2√3 cm

Where the marks are

  • Uses tangent ⟂ radius to get ∠PAB = 60° (1/2 mark)
  • Uses PA = PB to get ∠PBA = 60° (1/2 mark)
  • Shows ∠APB = 60°, so triangle ABP is equilateral (1/2 mark)
  • PA = 6 cm (1/2 mark)
  • Uses tan 30° = OA/PA in triangle OAP (1/2 mark)
  • OA = 2√3 cm (1/2 mark)

Revise the chapter

  • Chapter: Circles (notes, formulas, where marks are lost and every past question)
Practise this chapterBoard-style questions from Circles, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24 (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.