Perpendicular bisector of chord

Find the perpendicular bisector of a line segment (often a chord), which passes through the circle's centre.

10 questions · Moderate -0.6

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CAIE P1 2022 November Q1
6 marks Easy -1.2
1 Points \(A\) and \(B\) have coordinates \(( 5,2 )\) and \(( 10 , - 1 )\) respectively.
  1. Find the equation of the perpendicular bisector of \(A B\).
  2. Find the equation of the circle with centre \(A\) which passes through \(B\).
Edexcel C12 2014 January Q15
14 marks Moderate -0.8
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e878227b-d625-4ef2-ac49-a9dc05c5321a-40_883_824_212_568} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Diagram NOT drawn to scale The points \(X\) and \(Y\) have coordinates \(( 0,3 )\) and \(( 6,11 )\) respectively. \(X Y\) is a chord of a circle \(C\) with centre \(Z\), as shown in Figure 3.
  1. Find the gradient of \(X Y\). The point \(M\) is the midpoint of \(X Y\).
  2. Find an equation for the line which passes through \(Z\) and \(M\). Given that the \(y\) coordinate of \(Z\) is 10 ,
  3. find the \(x\) coordinate of \(Z\),
  4. find the equation of the circle \(C\), giving your answer in the form $$x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$$ where \(a\), \(b\) and \(c\) are constants.
Edexcel C2 2007 June Q7
9 marks Moderate -0.5
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{22ebc302-765c-4734-b312-b286ccb20be9-09_778_988_223_500} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The points \(A\) and \(B\) lie on a circle with centre \(P\), as shown in Figure 3.
The point \(A\) has coordinates \(( 1 , - 2 )\) and the mid-point \(M\) of \(A B\) has coordinates \(( 3,1 )\). The line \(l\) passes through the points \(M\) and \(P\).
  1. Find an equation for \(l\). Given that the \(x\)-coordinate of \(P\) is 6 ,
  2. use your answer to part (a) to show that the \(y\)-coordinate of \(P\) is - 1 ,
  3. find an equation for the circle.
OCR MEI C1 2015 June Q11
12 marks Moderate -0.5
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55e1f96-670a-4bc3-9e77-92d28769b7f5-3_700_751_906_641} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows a sketch of the circle with equation \(( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 125\) and centre C . The points \(\mathrm { A } , \mathrm { B }\), D and E are the intersections of the circle with the axes.
  1. Write down the radius of the circle and the coordinates of C .
  2. Verify that B is the point \(( 21,0 )\) and find the coordinates of \(\mathrm { A } , \mathrm { D }\) and E .
  3. Find the equation of the perpendicular bisector of BE and verify that this line passes through C .
OCR C1 Q10
12 marks Moderate -0.5
10.
\includegraphics[max width=\textwidth, alt={}, center]{af6fdbed-fcab-4db8-9cdf-fd049ce720fd-3_668_787_918_431} The diagram shows the circle \(C\) and the straight line \(l\).
The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).
  1. Find the gradient of 1 .
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the coordinates of the centre of \(C\).
  4. Show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0$$
OCR MEI C1 Q4
12 marks Moderate -0.5
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12a2563e-fce4-4c82-84aa-96603a50d6ad-2_520_1115_339_565} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).
  1. Find the equation of the line through \(\mathbf { A }\) and \(\mathbf { B }\).
  2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
  4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.
OCR MEI C1 Q1
12 marks Moderate -0.5
1 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01569a16-66ba-422e-a74d-6f9430dd245b-1_520_1122_357_551} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).
  1. Find the equation of the line through \(\mathbf { A }\) and \(\mathbf { B }\).
  2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
  4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.
OCR MEI C1 2010 June Q11
12 marks Moderate -0.3
11 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7942af14-fb10-42ba-b77a-b50ce65a7bcc-3_527_1125_794_513} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).
  1. Find the equation of the line through A and B .
  2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
  4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.
Edexcel C1 Q7
13 marks Moderate -0.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{beec3800-227a-45a1-87b8-0ea96e0e6270-4_661_746_283_721}
\end{figure} The points \(A\) and \(B\) have coordinates \(( 2 , - 3 )\) and \(( 8,5 )\) respectively, and \(A B\) is a chord of a circle with centre \(C\), as shown in Fig. 1.
  1. Find the gradient of \(A B\). The point \(M\) is the mid-point of \(A B\).
  2. Find an equation for the line through \(C\) and \(M\).
    (5) Given that the \(x\)-coordinate of \(C\) is 4 ,
  3. find the \(y\)-coordinate of \(C\),
    (2)
  4. show that the radius of the circle is \(\frac { 5 \sqrt { } 17 } { 4 }\).
    (4)
Edexcel C2 Q8
12 marks Moderate -0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{857bf144-b03e-4b46-b043-1119b30f9e78-4_533_685_242_497} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the circle \(C\) and the straight line \(l\). The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).
  1. Find the gradient of \(l\).
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the coordinates of the centre of \(C\).
  4. Show that \(C\) has the equation \(x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0\).