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Circles, parabolas, ellipses and hyperbolas
5 ECAT Mathematics questions with answers and explanations.
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What are the centre and radius of the circle x^2 + y^2 - 6x + 8y = 0?
- 1 Centre (3, -4), radius 5
- 2 Centre (-3, 4), radius 5
- 3 Centre (3, -4), radius 25
- 4 Centre (6, -8), radius 10
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Correct answer: Centre (3, -4), radius 5
Completing the square gives (x - 3)^2 + (y + 4)^2 = 25, so the centre is (3, -4) and the radius is 5.
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What is the length of the major axis of the ellipse x^2 / 9 + y^2 / 4 = 1?
- 1 3
- 2 4
- 3 6
- 4 9
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Correct answer: 6
The larger denominator is 9, so a = 3 and the major axis has length 2a = 6.
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What are the asymptotes of the hyperbola x^2 - y^2 = 1?
- 1 y = 0 and x = 0
- 2 y = x and y = -x
- 3 y = 2x and y = -2x
- 4 x = 1 and x = -1
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Correct answer: y = x and y = -x
For x^2 - y^2 = 1 the asymptotes come from x^2 - y^2 = 0, which gives y = x and y = -x.
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What is the focus of the parabola x^2 = 12y?
- 1 (0, 3)
- 2 (3, 0)
- 3 (0, 12)
- 4 (12, 0)
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Correct answer: (0, 3)
Comparing with x^2 = 4ay gives 4a = 12, so a = 3 and the focus is (0, 3).
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What is the eccentricity of a parabola?
- 1 0
- 2 Less than 1
- 3 1
- 4 Greater than 1
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Correct answer: 1
A parabola is defined by its distance to the focus equalling its distance to the directrix, so e = 1.