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WMA12/01 Oct 2023, Question 7: Chords and Tangent Geometry

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Circle C has centre X(4, −3); line l is y = (5/2)x − 55/2 and touches C at N. Show the stated line through X and N, then find N and the circle equation.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Show that the line through X and N has equation 2x + 5y + 7 = 0, using that l is tangent at N.

  1. 1Gradient of l is 5/2, and XN is perpendicular to the tangent, so gradient of XN = −2/5
  2. 2Line through X(4, −3): y + 3 = −(2/5)(x − 4)
  3. 3Multiply by 5: 5y + 15 = −2x + 8
  4. 4So 2x + 5y + 7 = 0

AnswerShown: gradient of XN = −2/5, so y + 3 = −(2/5)(x − 4) ⇒ 5y + 15 = −2x + 8 ⇒ 2x + 5y + 7 = 0

Part (bi) · 2 marks

Find the coordinates of the tangency point N.

  1. 1Substitute y = (5/2)x − 55/2 into 2x + 5y + 7 = 0
  2. 22x + (25/2)x − 275/2 + 7 = 0 ⇒ 4x + 25x − 275 + 14 = 0 ⇒ 29x = 261
  3. 3x = 9, y = 45/2 − 55/2 = −5

AnswerN = (9, −5)

Part (bii) · 3 marks

Find an equation for circle C.

  1. 1Radius² = XN² = (9 − 4)² + (−5 + 3)²
  2. 2= 25 + 4 = 29
  3. 3C: (x − 4)² + (y + 3)² = 29

Answer(x − 4)² + (y + 3)² = 29

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Also searched as: WMA12 October 2023 · WMA12/01 · wma12 oct23 · IAS Maths Pure Mathematics 2 October 2023 · question 7.
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