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ESAT Maths 2 Coordinate geometry questions

8 original ESAT Maths 2 coordinate geometry questions in the real exam style, each with a step-by-step worked solution and an explanation of why the tempting wrong answers are wrong. Try each one before opening the solution.

Methods on this page: How two circles meetGreatest value of ax + by on a circle: the line just touching it

Question 1ESAT Maths 2Standard

Circle C1C_1 has equation x2+y2−2x−4y−59=0x^2 + y^2 - 2x - 4y - 59 = 0. Circle C2C_2 has centre (−5,10)(-5, 10) and radius rr, where rr is a positive integer. For how many values of rr do the circles meet at exactly two points?
  1. A88
  2. B1414
  3. C1515
  4. D1616
  5. E1717
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Answer: C · Method: How two circles meet

  1. C1C_1 has centre (1,2)(1, 2) and radius 88. The centres are 36+64=10\sqrt{36 + 64} = 10 apart.
  2. Two circles meet at two points exactly when the distance between centres is strictly between the difference and the sum of the radii: ∣r−8∣<10<r+8\lvert r - 8 \rvert < 10 < r + 8.
  3. So 2<r<182 < r < 18: that's 1515 integer values.

Why the tempting wrong answers are wrong

  • E: At r=2r = 2 and r=18r = 18 the circles only touch, at one point.

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Question 2ESAT Maths 2Standard

Circle C1C_1 has equation (x−3)2+(y+4)2=64(x - 3)^2 + (y + 4)^2 = 64. Circle C2C_2 has centre (−6,−16)(-6, -16) and radius rr, where rr is a positive integer. For how many values of rr do the circles meet at exactly two points?
  1. A88
  2. B1313
  3. C1414
  4. D1515
  5. E1616
  6. F1717
Show the answer and worked solution

Answer: D · Method: How two circles meet

  1. C1C_1 has centre (3,−4)(3, -4) and radius 88. The centres are 81+144=15\sqrt{81 + 144} = 15 apart.
  2. Two circles meet at two points exactly when the distance between centres is strictly between the difference and the sum of the radii: ∣r−8∣<15<r+8\lvert r - 8 \rvert < 15 < r + 8.
  3. So 7<r<237 < r < 23: that's 1515 integer values.

Why the tempting wrong answers are wrong

  • F: At r=7r = 7 and r=23r = 23 the circles only touch, at one point.

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Question 3ESAT Maths 2Standard

Circle C1C_1 has equation (x+1)2+(y−1)2=9(x + 1)^2 + (y - 1)^2 = 9. Circle C2C_2 has centre (7,7)(7, 7) and radius rr, where rr is a positive integer. For how many values of rr do the circles meet at exactly two points?
  1. A33
  2. B44
  3. C55
  4. D66
  5. E77
  6. F88
  7. G99
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Answer: C · Method: How two circles meet

  1. C1C_1 has centre (−1,1)(-1, 1) and radius 33. The centres are 64+36=10\sqrt{64 + 36} = 10 apart.
  2. Two circles meet at two points exactly when the distance between centres is strictly between the difference and the sum of the radii: ∣r−3∣<10<r+3\lvert r - 3 \rvert < 10 < r + 3.
  3. So 7<r<137 < r < 13: that's 55 integer values.

Why the tempting wrong answers are wrong

  • E: At r=7r = 7 and r=13r = 13 the circles only touch, at one point.

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Question 4ESAT Maths 2Standard

Circle C1C_1 has equation x2+y2+4x−8y−44=0x^2 + y^2 + 4x - 8y - 44 = 0. Circle C2C_2 has centre (6,10)(6, 10) and radius rr, where rr is a positive integer. For how many values of rr do the circles meet at exactly two points?
  1. A88
  2. B1313
  3. C1414
  4. D1515
  5. E1616
  6. F1717
Show the answer and worked solution

Answer: D · Method: How two circles meet

  1. C1C_1 has centre (−2,4)(-2, 4) and radius 88. The centres are 64+36=10\sqrt{64 + 36} = 10 apart.
  2. Two circles meet at two points exactly when the distance between centres is strictly between the difference and the sum of the radii: ∣r−8∣<10<r+8\lvert r - 8 \rvert < 10 < r + 8.
  3. So 2<r<182 < r < 18: that's 1515 integer values.

Why the tempting wrong answers are wrong

  • F: At r=2r = 2 and r=18r = 18 the circles only touch, at one point.

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Question 5ESAT Maths 2Standard

Circle C1C_1 has equation x2+y2+8x+4y−44=0x^2 + y^2 + 8x + 4y - 44 = 0. Circle C2C_2 has centre (−12,4)(-12, 4) and radius rr. There are two values of rr for which the circles touch each other. What is the product of these two values?
  1. A3535
  2. B3636
  3. C3737
  4. D8080
  5. E100100
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Answer: B · Method: How two circles meet

  1. C1C_1 has centre (−4,−2)(-4, -2) and radius 88. The distance between the centres is 1010.
  2. They touch externally when r+8=10r + 8 = 10, so r=2r = 2, and internally when r−8=10r - 8 = 10, so r=18r = 18.
  3. The product is 3636.

Why the tempting wrong answers are wrong

  • E: One value is when they touch externally (r+r1=dr + r_1 = d) and the other internally (r−r1=dr - r_1 = d).

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Question 6ESAT Maths 2Standard

Circle C1C_1 has equation (x−3)2+(y−2)2=4(x - 3)^2 + (y - 2)^2 = 4. Circle C2C_2 has centre (6,−2)(6, -2) and radius rr. There are two values of rr for which the circles touch each other. What is the product of these two values?
  1. A1010
  2. B2020
  3. C2121
  4. D2222
  5. E2323
  6. F2525
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Answer: C · Method: How two circles meet

  1. C1C_1 has centre (3,2)(3, 2) and radius 22. The distance between the centres is 55.
  2. They touch externally when r+2=5r + 2 = 5, so r=3r = 3, and internally when r−2=5r - 2 = 5, so r=7r = 7.
  3. The product is 2121.

Why the tempting wrong answers are wrong

  • F: One value is when they touch externally (r+r1=dr + r_1 = d) and the other internally (r−r1=dr - r_1 = d).

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Question 7ESAT Maths 2Challenging

Given that x2+y2=4x^2 + y^2 = 4, what is the greatest possible value of 2x+y2x + y?
  1. A5\sqrt{5}
  2. B33
  3. C252\sqrt{5}
  4. D22\sqrt{22}
  5. E262\sqrt{6}
  6. F55
  7. G66
  8. H454\sqrt{5}
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Answer: C · Method: Greatest value of ax + by on a circle: the line just touching it

  1. The greatest value is where the line 2x+y=c2x + y = c just touches the circle, i.e. where its distance from the origin equals the radius 22.
  2. Distance =c22+12= \dfrac{c}{\sqrt{2^2 + 1^2}}, so c=2×5=25c = 2 \times \sqrt{5} = 2\sqrt{5}.

Why the tempting wrong answers are wrong

  • A: That's for the unit circle. Scale by the radius.
  • G: xx and yy can't both reach their largest values at the same point of the circle.

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Question 8ESAT Maths 2Challenging

Given that x2+y2=9x^2 + y^2 = 9, what is the greatest possible value of 2x+3y2x + 3y?
  1. A13\sqrt{13}
  2. B55
  3. C3133\sqrt{13}
  4. D3143\sqrt{14}
  5. E1515
  6. F6136\sqrt{13}
  7. G9139\sqrt{13}
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Answer: C · Method: Greatest value of ax + by on a circle: the line just touching it

  1. The greatest value is where the line 2x+3y=c2x + 3y = c just touches the circle, i.e. where its distance from the origin equals the radius 33.
  2. Distance =c22+32= \dfrac{c}{\sqrt{2^2 + 3^2}}, so c=3×13=313c = 3 \times \sqrt{13} = 3\sqrt{13}.

Why the tempting wrong answers are wrong

  • A: That's for the unit circle. Scale by the radius.
  • E: xx and yy can't both reach their largest values at the same point of the circle.

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