Lines Meeting Circles: Tangents and Chords · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Lines Meeting Circles: Tangents and Chords

Mathematics · Geometry · ages 16-17
Name ______________________   Date ____________

  1. A line that misses a circle gives a quadratic with two real roots.

    Circle one:   True   False

  2. What is the discriminant of 3x² - 4x + 2 = 0?

    Answer: ______________

  3. x^2 + x - 12 = 0 gives the crossing x values of y = x + 1 with x^2 + y^2 = 25. What is the larger one?

    Answer: ______________

  4. A straight line crosses a circle at two points. What is the piece of that line lying between the two points called?

    • a chord
    • a tangent
    • a radius
    • a diameter

  5. Line y = x + 1 meets circle x^2 + y^2 = 25 at (3, 4) and (-4, -3). How many points is that?

    • 1
    • 0
    • 2
    • 4

  6. A line circle substitution gives discriminant 49. How many intersection points?

    • 1
    • 0
    • 49
    • 2

  7. The line y = x + c is a tangent to the circle x² + y² = 8. What is the positive value of c?

    Answer: ______________

  8. Find the tangent to the circle x² + y² = 100 at the point (8, 6).

    • 3x + 4y - 48 = 0
    • 4x - 3y - 14 = 0
    • 4x + 3y - 50 = 0
    • 3x + 4y - 50 = 0

  9. The vertical line x = k is a tangent to the circle (x - 3)² + (y - 2)² = 16, touching it on the right-hand side. What is k?

    Answer: ______________

  10. Find the tangent to the circle (x - 2)² + (y + 1)² = 169 at the point (14, 4).

    • 12x + 5y - 188 = 0
    • 5x + 12y - 118 = 0
    • 12x - 5y - 148 = 0
    • 12x + 5y - 168 = 0
LightMySky · lightmysky.comW1-mt_ELA4IzIVdv-s1

Answer key

For grown-ups. Fold this page away before handing over the rest.

Lines Meeting Circles: Tangents and Chords W1-mt_ELA4IzIVdv-s1

  1. False · A real root is a meeting point. A line that misses has no meeting points, so its quadratic has no real roots and a negative discriminant.
  2. -8 · With a = 3, b = -4 and c = 2, b² - 4ac is 16 - 24, which is -8.
  3. 3 · Factors (x+4)(x-3): roots -4 and 3.
  4. a chord · The piece of a crossing line that lies inside the circle is a chord. A tangent meets the circle only once.
  5. 2 · Two listed pairs. The 4 option counts coordinates, not points.
  6. 2 · Positive discriminant means two crossings. The 1 option is the zero case.
  7. 4 · Substituting gives 2x² + 2cx + c² - 8 = 0. Its discriminant is 4c² - 8(c² - 8) = 64 - 4c², and setting that to zero gives c² = 16, so c = 4 or -4.
  8. 4x + 3y - 50 = 0 · The radius runs from (0, 0) to (8, 6), so its gradient is 6/8 = 3/4. The tangent gradient is -4/3, and y - 6 = -(4/3)(x - 8) gives 4x + 3y - 50 = 0.
  9. 7 · The centre is (3, 2) and the radius is 4. A vertical tangent on the right touches one radius to the right of the centre, at x = 3 + 4 = 7.
  10. 12x + 5y - 188 = 0 · The centre is (2, -1), so the radius gradient is (4 - (-1)) / (14 - 2) = 5/12. The tangent gradient is -12/5, and y - 4 = -(12/5)(x - 14) gives 12x + 5y - 188 = 0.
Worksheet · LightMySky