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Circles, parabolas, ellipses and hyperbolas

Written by Hassan Raza (Mathematics)

ECAT · Mathematics · 5 questions

Name: ____________________   Date: ____________

  1. What are the centre and radius of the circle x² + y² - 6x + 8y = 0?

    • A Centre (-3, 4), radius 5
    • B Centre (6, -8), radius 10
    • C Centre (3, -4), radius 5
    • D Centre (3, -4), radius 25

  2. What is the length of the major axis of the ellipse x² / 9 + y² / 4 = 1?

    • A 4
    • B 3
    • C 9
    • D 6

  3. What are the asymptotes of the hyperbola x² - y² = 1?

    • A y = 0 and x = 0
    • B y = x and y = -x
    • C x = 1 and x = -1
    • D y = 2x and y = -2x

  4. What is the focus of the parabola x² = 12y?

    • A (3, 0)
    • B (12, 0)
    • C (0, 3)
    • D (0, 12)

  5. What is the eccentricity of a parabola?

    • A Less than 1
    • B 0
    • C 1
    • D Greater than 1

Answer key

1. C, 2. D, 3. B, 4. C, 5. C

  1. Completing the square gives (x - 3)² + (y + 4)² = 25, so the centre is (3, -4) and the radius is 5.

  2. The larger denominator is 9, so a = 3 and the major axis has length 2a = 6.

  3. For x² - y² = 1 the asymptotes come from x² - y² = 0, which gives y = x and y = -x.

  4. Comparing with x² = 4ay gives 4a = 12, so a = 3 and the focus is (0, 3).

  5. A parabola is defined by its distance to the focus equalling its distance to the directrix, so e = 1.