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WMA12/01 Jan 2023, Question 6: Circles Meeting Lines, and Tangents

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 10 marks

The question, in short

For circle C with equation x² + y² + 8x − 4y = 0, determine its centre and exact radius, then use the tangent x + 2y + 10 = 0 at point P to find P and the normal at P.

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

Part (ai) · 1 mark

Work out the centre coordinates of circle C from its expanded equation.

  1. 1Complete the square: (x + 4)² − 16 + (y − 2)² − 4 = 0
  2. 2Centre (−4, 2)

Answer(−4, 2)

Part (aii) · 2 marks

Calculate the exact radius of circle C.

  1. 1(x + 4)² + (y − 2)² = 20
  2. 2r = √20 = 2√5

Answerr = 2√5

Part (b) · 4 marks

Using the given tangent line at P, determine the coordinates of point P on C.

  1. 1From the tangent: x = −2y − 10
  2. 2(−2y − 10)² + y² + 8(−2y − 10) − 4y = 0
  3. 34y² + 40y + 100 + y² − 16y − 80 − 4y = 0 ⇒ 5y² + 20y + 20 = 0
  4. 4y² + 4y + 4 = 0 ⇒ (y + 2)² = 0 ⇒ y = −2
  5. 5x = −2(−2) − 10 = −6, so P(−6, −2)

AnswerP(−6, −2)

Part (c) · 3 marks

Give the normal line to C at P as y = mx + c with integer m and c.

  1. 1Tangent gradient: x + 2y + 10 = 0 ⇒ m = −½
  2. 2Normal gradient = 2
  3. 3Through P(−6, −2): y + 2 = 2(x + 6)
  4. 4y = 2x + 10

Answery = 2x + 10

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Also searched as: WMA12 January 2023 · WMA12/01 · wma12 jan23 · IAS Maths Pure Mathematics 2 January 2023 · question 6.
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