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PT is a tangent to the circle… (Class 10 Maths 2023, Q22)

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MathsCBSE Class 10 · Standard · Board exam 2023, set 30/5/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2023: question 22

Set in the CBSE Class 10 board exam of 2023, paper 30/5/1, section B, question 22, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q22 · 2 marks · Very short answerCircles

In the given figure, PT is a tangent to the circle centred at O. OC is perpendicular to chord AB. Prove that .

Figure in the paper: Circle with centre O. P is outside the circle; PT is a tangent touching the circle at T. Line from P cuts the circle at A and B (B further along the chord). OC is drawn from O perpendicular to chord AB, with C on AB between A and B. OP is also drawn. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/5/1.

Hints

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

Hint 1

Write PA and PB as sums or differences of lengths along the line P–A–C–B.

Hint 2

What does a perpendicular from the centre do to a chord?

Hint 3

After replacing CB by AC, use the identity .

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. From the figure, A lies between P and C, and C lies between A and B. So and .
    Hence .

    1 mark

  2. , and the perpendicular from the centre to a chord bisects the chord, so .
    .

    ½ mark

  3. [using ]. Hence proved.

    ½ mark

Answer: PA · PB = (PC − AC)(PC + AC) = PC² − AC² (proved).

Where the marks are

  • Writes PA = PC − AC and PB = PC + CB, so PA·PB = (PC − AC)(PC + CB) (1 mark)
  • Uses AC = CB since the perpendicular from the centre bisects the chord (½ mark)
  • Simplifies to PC² − AC² (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 paper2023 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.