Range of parameter for intersection

Find the range of values of a parameter (often k or m) for which a line or point satisfies certain intersection conditions with a circle.

8 questions · Standard +0.3

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CAIE P1 2022 June Q9
10 marks Standard +0.3
9 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 26 = 0\).
  1. Find the coordinates of the centre of the circle and the radius. Hence find the coordinates of the lowest point on the circle.
  2. Find the set of values of the constant \(k\) for which the line with equation \(y = k x - 5\) intersects the circle at two distinct points.
Edexcel P2 2020 January Q6
8 marks Standard +0.3
6. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 14 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\),
    2. the exact radius of \(C\). The line with equation \(y = k\), where \(k\) is a constant, is a tangent to \(C\).
  2. Find the possible values of \(k\). The line with equation \(y = p\), where \(p\) is a negative constant, is a chord of \(C\).
    Given that the length of this chord is 4 units,
  3. find the value of \(p\).
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OCR H240/03 2019 June Q2
8 marks Standard +0.3
2 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 4 = 0\).
  1. Find
    1. the coordinates of \(C\),
    2. the radius of the circle.
  2. Determine the set of values of \(k\) for which the line \(y = k x - 3\) does not intersect or touch the circle.
Edexcel AS Paper 1 2018 June Q14
9 marks Standard +0.3
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\)
    2. the radius of \(C\) The line with equation \(y = k x\), where \(k\) is a constant, cuts \(C\) at two distinct points.
  2. Find the range of values for \(k\).
Edexcel AS Paper 1 2023 June Q6
5 marks Standard +0.3
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + k = 0$$ where \(k\) is a constant.
  1. Find the coordinates of the centre of \(C\). Given that \(C\) does not cut or touch the \(x\)-axis,
  2. find the range of possible values for \(k\).
Edexcel Paper 1 2023 June Q10
9 marks Standard +0.8
  1. A circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } + 6 k x - 2 k y + 7 = 0$$ where \(k\) is a constant.
  1. Find in terms of \(k\),
    1. the coordinates of the centre of \(C\)
    2. the radius of \(C\) The line with equation \(y = 2 x - 1\) intersects \(C\) at 2 distinct points.
  2. Find the range of possible values of \(k\).
Edexcel Paper 1 Specimen Q3
4 marks Moderate -0.8
3. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 4 x + 10 y = k$$ where \(k\) is a constant.
  1. Find the coordinates of the centre of \(C\).
  2. State the range of possible values for \(k\).
AQA Paper 2 2021 June Q7
8 marks Standard +0.8
7 A circle has equation $$x ^ { 2 } + y ^ { 2 } - 6 x - 8 y = p$$ 7
    1. State the coordinates of the centre of the circle.
      7
  2. (ii) Find the radius of the circle in terms of \(p\).
    7
  3. The circle intersects the coordinate axes at exactly three points. Find the two possible values of \(p\).