3.9 The equation of a tangent at a point on a circle Quiz: AQA Further Maths, Unit 3

The 20 questions

1. A tangent to a circle is perpendicular to which line at the point of contact?

   • The x-axis
   • A secant
   • The radius
   • A chord

2. What is the angle between a tangent and the radius at the point of contact?

   • 90 degrees
   • 180 degrees
   • 60 degrees
   • 45 degrees

3. What is the gradient of the radius from the origin to (-2, 5) on a circle centred at the origin?

   • -5/2
   • 2/5
   • -2/5
   • 5/2

4. A circle x^2 + y^2 = 25 has the tangent at (3, 4). Which is its equation?

   • 3x + 4y = 25
   • 3x - 4y = 25
   • 3x + 4y = 12
   • 4x + 3y = 25

5. Find the tangent to x^2 + y^2 = 10 at (1, 3).

   • x - 3y = -8
   • x + 3y = 9
   • x + 3y = 10
   • 3x + y = 6

6. Find the tangent to x^2 + y^2 = 13 at (2, 3).

   • 2x + 3y = 6
   • 3x + 2y = 13
   • 2x + 3y = 13
   • 2x - 3y = 13

7. The circle (x - 2)^2 + (y - 1)^2 = 10 has the point (5, 2) on it. What is the gradient of the tangent there?

   • 1/3
   • -1/3
   • 3
   • -3

8. What is the relationship between lengths of two tangents drawn from the same point to a circle?

   • One is double
   • They are perpendicular
   • They are equal
   • They are parallel

9. A chord of length 8 lies in a circle of radius 5. How far is the chord from the centre?

   • sqrt(41)
   • 4
   • 5
   • 3

10. On the circle x^2 + y^2 = 25, what is the gradient of the tangent at (-4, 3)?

    • -4/3
    • -3/4
    • 4/3
    • 3/4

11. The point (5, 0) lies on (x - 1)^2 + (y + 2)^2 = 20. What is the gradient of the tangent there?

    • -1/2
    • 1/2
    • 2
    • -2

12. Where does the tangent at (3, 4) to x^2 + y^2 = 25 meet the x-axis?

    • x = 25/3
    • x = 3
    • x = 5
    • x = 25/4

13. What is the equation of the tangent to x^2 + y^2 = 25 at (0, 5)?

    • y = -5
    • y = 5
    • y = 0
    • x = 5

14. What angle does a tangent to a circle make with the radius at the contact point?

    • 180 degrees
    • 90 degrees
    • 0 degrees
    • 45 degrees

15. The tangent at (p, q) to x^2 + y^2 = 13 is 2x - 3y = 13. Which is the point of contact?

    • (2, -3)
    • (-2, 3)
    • (3, 2)
    • (3, -2)

16. Find the tangent at (4, 2) to (x - 1)^2 + (y - 1)^2 = 10.

    • 3x + y = 14
    • 3x - y = 10
    • x + 3y = 10
    • 3x + y = 10

17. Does the tangent line 4x + 2y = 20 pass through the point (0, 10)?

    • Yes
    • Only if x = 2
    • No
    • Cannot be determined

18. How many points does the line y = 5 share with the circle x^2 + y^2 = 25?

    • One point
    • No points
    • Infinitely many points
    • Two points

19. The tangent at (6, 8) to x^2 + y^2 = 100 meets the y-axis at which point?

    • (0, 12.5)
    • (0, 10)
    • (0, 8)
    • (0, 12)

20. A circle x^2 + y^2 = 17 contains the point (1, 4). What is the equation of the tangent there?

    • x + 4y = 4
    • x + 4y = 17
    • x - 4y = -15
    • 4x - y = 0

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