Chord length calculation

Find the length of a chord in a circle, typically using the distance formula between intersection points or perpendicular distance from centre.

13 questions · Standard +0.2

1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle
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CAIE P1 2021 November Q7
9 marks Standard +0.3
7 A circle with centre \(( 5,2 )\) passes through the point \(( 7,5 )\).
  1. Find an equation of the circle.
    The line \(y = 5 x - 10\) intersects the circle at \(A\) and \(B\).
  2. Find the exact length of the chord \(A B\).
Edexcel C12 Specimen Q12
11 marks Standard +0.3
12. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\)
  1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
  2. Find an equation for \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
    Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
  3. find the length of \(P Q\), giving your answer in its simplest surd form.
Edexcel C2 2010 January Q8
12 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e3faf018-37a8-48ef-b100-81402a8ec87f-11_1262_1178_203_386} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the circle \(C\) with centre \(N\) and equation $$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = \frac { 169 } { 4 }$$
  1. Write down the coordinates of \(N\).
  2. Find the radius of \(C\). The chord \(A B\) of \(C\) is parallel to the \(x\)-axis, lies below the \(x\)-axis and is of length 12 units as shown in Figure 3.
  3. Find the coordinates of \(A\) and the coordinates of \(B\).
  4. Show that angle \(A N B = 134.8 ^ { \circ }\), to the nearest 0.1 of a degree. The tangents to \(C\) at the points \(A\) and \(B\) meet at the point \(P\).
  5. Find the length \(A P\), giving your answer to 3 significant figures.
Edexcel C2 2010 June Q10
11 marks Standard +0.3
10. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\).
  1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
  2. Find an equation for \(l _ { 1 }\). The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
    Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
  3. find the length of \(P Q\), giving your answer in its simplest surd form. \includegraphics[max width=\textwidth, alt={}, center]{571780c2-945b-4636-b7c3-0bd558d28710-15_115_127_2461_1814}
OCR C1 Q9
10 marks Moderate -0.3
9. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0$$
  1. Find the coordinates of the centre of \(C\).
  2. Find the radius of \(C\).
  3. Sketch \(C\). Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
  4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).
OCR MEI C1 Q3
12 marks Standard +0.3
3 The circle \(( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\) has centre C.
  1. Write down the radius of the circle and the coordinates of C .
  2. Find the coordinates of the intersections of the circle with the \(x\) - and \(y\)-axes.
  3. Show that the points \(\mathrm { A } ( 1,6 )\) and \(\mathrm { B } ( 7,4 )\) lie on the circle. Find the coordinates of the midpoint of AB . Find also the distance of the chord AB from the centre of the circle.
OCR MEI AS Paper 1 2024 June Q8
10 marks Standard +0.3
8 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 16 y + 48 = 0\).
  1. Find the coordinates of C . A line has equation \(\mathrm { y } = \mathrm { x } - 2\) and intersects the circle at the points A and B . The midpoints of AC and BC are \(\mathrm { A } ^ { \prime }\) and \(\mathrm { B } ^ { \prime }\) respectively.
  2. Determine the exact distance \(\mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime }\).
AQA C1 2015 June Q4
11 marks Standard +0.3
  1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = d$$
    1. State the coordinates of \(C\).
    2. Find the radius of the circle, giving your answer in the form \(n \sqrt { 2 }\).
  3. The point \(P\) with coordinates \(( 4 , k )\) lies on the circle. Find the possible values of \(k\).
  4. The points \(Q\) and \(R\) also lie on the circle, and the length of the chord \(Q R\) is 2 . Calculate the shortest distance from \(C\) to the chord \(Q R\).
    [0pt] [2 marks]
Edexcel C2 Q6
7 marks Standard +0.3
The circle \(C\), with centre \(A\), has equation $$x^2 + y^2 - 6x + 4y - 12 = 0.$$
  1. Find the coordinates of \(A\). [2]
  2. Show that the radius of \(C\) is 5. [2]
The points \(P\), \(Q\) and \(R\) lie on \(C\). The length of \(PQ\) is 10 and the length of \(PR\) is 3.
  1. Find the length of \(QR\), giving your answer to 1 decimal place. [3]
OCR MEI C1 2013 June Q10
12 marks Moderate -0.8
The circle \((x - 3)^2 + (y - 2)^2 = 20\) has centre C.
  1. Write down the radius of the circle and the coordinates of C. [2]
  2. Find the coordinates of the intersections of the circle with the \(x\)- and \(y\)-axes. [5]
  3. Show that the points A\((1, 6)\) and B\((7, 4)\) lie on the circle. Find the coordinates of the midpoint of AB. Find also the distance of the chord AB from the centre of the circle. [5]
OCR H240/02 2023 June Q6
10 marks Standard +0.3
A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_6} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.
  1. In this question you must show detailed reasoning. Show that the area of triangle \(ABC\) is \(\frac{5}{2}\sqrt{9 - 2a}\). [7]
    1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
    2. Give a geometrical interpretation of the case in part (b)(i). [1]
  3. Give a geometrical interpretation of the case where \(a = 5\). [1]
AQA Paper 1 Specimen Q11
8 marks Moderate -0.3
A circle with centre \(C\) has equation \(x^2 + y^2 + 8x - 12y = 12\)
  1. Find the coordinates of \(C\) and the radius of the circle. [3 marks]
  2. The points \(P\) and \(Q\) lie on the circle. The origin is the midpoint of the chord \(PQ\). Show that \(PQ\) has length \(n\sqrt{3}\), where \(n\) is an integer. [5 marks]
SPS SPS FM 2024 October Q9
9 marks Standard +0.8
A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.
  1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
    1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
    2. Give a geometrical interpretation of the case in part (b)(i). [1]
  3. Give a geometrical interpretation of the case where \(a = 5\). [1]