Distance from centre to line

Calculate the perpendicular distance from the circle's centre to a given line, often to determine intersection properties.

5 questions · Moderate -0.1

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OCR C1 Q8
9 marks Standard +0.3
8.
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The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x - 18 y + 73 = 0\) and the straight line with equation \(y = 2 x - 3\).
  1. Find the coordinates of the centre and the radius of the circle.
  2. Find the coordinates of the point on the line which is closest to the circle.
Edexcel AS Paper 1 2022 June Q11
8 marks Moderate -0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d31369fa-9532-4a09-b67d-a3a3cbf7d586-34_833_1033_248_516} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the circle \(C\) with equation $$x ^ { 2 } + y ^ { 2 } - 10 x - 8 y + 32 = 0$$ and the line \(l\) with equation $$2 y + x + 6 = 0$$
  1. Find
    1. the coordinates of the centre of \(C\),
    2. the radius of \(C\).
  2. Find the shortest distance between \(C\) and \(l\).
OCR PURE 2020 October Q6
6 marks Moderate -0.3
6 In this question you must show detailed reasoning.
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The diagram shows the line \(3 y + x = 7\) which is a tangent to a circle with centre \(( 3 , - 2 )\).
Find an equation for the circle.
OCR Pure 1 2018 September Q7
11 marks Moderate -0.3
7 A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).
  1. Show that the line does not meet the circle.
  2. (a) Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
    (b) Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.
AQA Paper 1 2022 June Q8
11 marks Standard +0.3
8 The lines \(L _ { 1 }\) and \(L _ { 2 }\) are parallel.
\(L _ { 1 }\) has equation $$5 x + 3 y = 15$$ and \(L _ { 2 }\) has equation $$5 x + 3 y = 83$$ \(L _ { 1 }\) intersects the \(y\)-axis at the point \(P\).
The point \(Q\) is the point on \(L _ { 2 }\) closest to \(P\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-10_849_917_945_561} 8
    1. Find the coordinates of \(Q\).
      8
  2. (ii) Hence show that \(P Q = k \sqrt { 34 }\), where \(k\) is an integer to be found. 8
  3. A circle, \(C\), has centre ( \(a , - 17\) ).
    \(L _ { 1 }\) and \(L _ { 2 }\) are both tangents to \(C\).
    8
    1. Find \(a\).
      8
  5. (ii) Find the equation of \(C\).
    \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-13_2493_1732_214_139}