← Lesson📄 Worksheet✅ Memo📝 Exam test🗝️ Marking guide
∑ DANEMATHICS
FREE CAPS MATHS RESOURCES
Grade 12 · Analytical Geometry
The Circle & Its Tangent (Grade 12)
MEMORANDUM
Name: 
Class: 
Date: 
Mark
  / 16

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

  1. 1
    Write down the centre and radius of (x − 2)² + (y + 3)² = 25.
    (2)
    −4−22468−8−6−4−22Mxy
    Centre (2 ; −3), radius 5✓✓ (2)
  2. 2
    Determine the centre and radius of x² + y² − 4x + 6y − 12 = 0.
    (2)
    M(2 ; -3)P(6 ; 0)xy
    Completing the square: (x−2)² + (y+3)² = 25 → centre (2 ; −3), r = 5✓✓ (2)
  3. 3
    Show that P(6 ; 0) lies on the circle x² + y² − 4x + 6y − 12 = 0.
    (2)
    M(2 ; -3)P(6 ; 0)xy
    (6−2)² + (0+3)² = 16 + 9 = 25 = r², so P lies on the circle.✓✓ (2)
  4. 4
    Determine the gradient of the radius MP where M(2 ; −3) and P(6 ; 0).
    (2)
    −4−22468−8−6−4−22M(2 ; −3)P(6 ; 0)xy
    34✓✓ (2)
  5. 5
    Determine the equation of the tangent to the circle at P(6 ; 0).
    (2)
    M(2 ; -3)P(6 ; 0)xy
    m = −43 → y = −43x + 8✓✓ (2)
  6. 6
    Write down the equation of a circle with centre (0 ; 0) and radius 7.
    (2)
    −8−6−4−22468−8−6−4−22468M(0 ; 0)xy
    x² + y² = 49✓✓ (2)
  7. 7
    Determine the radius of x² + y² + 2x − 8y + 8 = 0.
    (2)
    −5−4−3−2−1123−112345678Mxy
    (x+1)² + (y−4)² = −8 + 1 + 16 = 9 → r = 3✓✓ (2)
  8. 8
    What is the relationship between a tangent and the radius at the point of contact?
    (2)
    M(2 ; -3)P(6 ; 0)xy
    They are perpendicular.✓✓ (2)
TOTAL: 16 marks
© Danemathics — free to print and share for classroom use.danemathics.com

Printing this sheet

  1. In the print dialog open More settings and untick “Headers and footers” — that is what prints the date, time and web address in the margin.
  2. Leave Pages per sheet on 1 so learners keep the full writing space.
  3. Choose Save as PDF as the destination to download it instead of printing.