Perpendicular line through point

Find the equation of a line perpendicular to a given line and passing through a specified point.

48 questions · Moderate -0.7

1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships
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CAIE P1 2003 June Q7
8 marks Moderate -0.8
7 The line \(L _ { 1 }\) has equation \(2 x + y = 8\). The line \(L _ { 2 }\) passes through the point \(A ( 7,4 )\) and is perpendicular to \(L _ { 1 }\).
  1. Find the equation of \(L _ { 2 }\).
  2. Given that the lines \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(B\), find the length of \(A B\).
CAIE P1 2013 November Q3
5 marks Moderate -0.8
3 The point \(A\) has coordinates \(( 3,1 )\) and the point \(B\) has coordinates \(( - 21,11 )\). The point \(C\) is the mid-point of \(A B\).
  1. Find the equation of the line through \(A\) that is perpendicular to \(y = 2 x - 7\).
  2. Find the distance \(A C\).
CAIE P1 2018 November Q4
6 marks Moderate -0.8
4 Two points \(A\) and \(B\) have coordinates \(( - 1,1 )\) and \(( 3,4 )\) respectively. The line \(B C\) is perpendicular to \(A B\) and intersects the \(x\)-axis at \(C\).
  1. Find the equation of \(B C\) and the \(x\)-coordinate of \(C\).
  2. Find the distance \(A C\), giving your answer correct to 3 decimal places.
Edexcel P1 2019 January Q3
5 marks Moderate -0.8
  1. The line \(l _ { 1 }\) has equation \(3 x + 5 y - 7 = 0\)
    1. Find the gradient of \(l _ { 1 }\)
    The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 6 , - 2 )\).
  2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C12 2017 January Q6
11 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-08_906_1100_127_388} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line \(l _ { 1 }\) has equation \(2 y = 3 x + 5\) The line \(l _ { 1 }\) cuts the \(x\)-axis at the point \(A\), as shown in Figure 2.
    1. State the gradient of \(l _ { 1 }\)
    2. Write down the \(x\) coordinate of point \(A\). Another straight line \(l _ { 2 }\) intersects \(l _ { 1 }\) at the point \(B\) with \(x\) coordinate 1 and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\)
  2. find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers,
  3. find the exact area of triangle \(A B C\).
Edexcel C12 2018 June Q3
6 marks Moderate -0.8
3. The line \(l _ { 1 }\) passes through the points \(A ( - 1,4 )\) and \(B ( 5 , - 8 )\)
  1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is perpendicular to the line \(l _ { 1 }\) and passes through the point \(B ( 5 , - 8 )\)
  2. Find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
    II
    "
Edexcel C1 2006 January Q3
5 marks Moderate -0.8
3. The line \(L\) has equation \(y = 5 - 2 x\).
  1. Show that the point \(P ( 3 , - 1 )\) lies on \(L\).
  2. Find an equation of the line perpendicular to \(L\), which passes through \(P\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C1 2011 January Q9
11 marks Moderate -0.8
9. The line \(L _ { 1 }\) has equation \(2 y - 3 x - k = 0\), where \(k\) is a constant. Given that the point \(A ( 1,4 )\) lies on \(L _ { 1 }\), find
  1. the value of \(k\),
  2. the gradient of \(L _ { 1 }\). The line \(L _ { 2 }\) passes through \(A\) and is perpendicular to \(L _ { 1 }\).
  3. Find an equation of \(L _ { 2 }\) giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The line \(L _ { 2 }\) crosses the \(x\)-axis at the point \(B\).
  4. Find the coordinates of \(B\).
  5. Find the exact length of \(A B\).
Edexcel C1 2012 January Q6
8 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff1cdb91-0286-4bc8-9e67-451500b2bf74-07_647_927_274_513} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The line \(l _ { 1 }\) has equation \(2 x - 3 y + 12 = 0\)
  1. Find the gradient of \(l _ { 1 }\). The line \(l _ { 1 }\) crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\), as shown in Figure 1. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(B\).
  2. Find an equation of \(l _ { 2 }\). The line \(l _ { 2 }\) crosses the \(x\)-axis at the point \(C\).
  3. Find the area of triangle \(A B C\).
Edexcel C1 2013 January Q5
7 marks Moderate -0.8
5. The line \(l _ { 1 }\) has equation \(y = - 2 x + 3\) The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 5,6 )\).
  1. Find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(l _ { 2 }\) crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\).
  2. Find the \(x\)-coordinate of \(A\) and the \(y\)-coordinate of \(B\). Given that \(O\) is the origin,
  3. find the area of the triangle \(O A B\).
Edexcel C1 2006 June Q11
15 marks Moderate -0.3
  1. The line \(l _ { 1 }\) passes through the points \(P ( - 1,2 )\) and \(Q ( 11,8 )\).
    1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
    The line \(l _ { 2 }\) passes through the point \(R ( 10,0 )\) and is perpendicular to \(l _ { 1 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(S\).
  2. Calculate the coordinates of \(S\).
  3. Show that the length of \(R S\) is \(3 \sqrt { 5 }\).
  4. Hence, or otherwise, find the exact area of triangle \(P Q R\).
Edexcel C1 2008 June Q10
13 marks Moderate -0.8
10.
[diagram]
The points \(Q ( 1,3 )\) and \(R ( 7,0 )\) lie on the line \(l _ { 1 }\), as shown in Figure 2.
The length of \(Q R\) is \(a \sqrt { } 5\).
  1. Find the value of \(a\). The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\), passes through \(Q\) and crosses the \(y\)-axis at the point \(P\), as shown in Figure 2. Find
  2. an equation for \(l _ { 2 }\),
  3. the coordinates of \(P\),
  4. the area of \(\triangle P Q R\).
Edexcel C1 2009 June Q8
8 marks Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e72d0d82-af0e-4f36-8446-a67b764fd7f3-09_908_1043_201_495} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(A\) and \(B\) have coordinates \(( 6,7 )\) and \(( 8,2 )\) respectively.
The line \(l\) passes through the point \(A\) and is perpendicular to the line \(A B\), as shown in Figure 1.
  1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. Given that \(l\) intersects the \(y\)-axis at the point \(C\), find
  2. the coordinates of \(C\),
  3. the area of \(\triangle O C B\), where \(O\) is the origin.
Edexcel C1 2012 June Q9
15 marks Standard +0.3
9. The line \(L _ { 1 }\) has equation \(4 y + 3 = 2 x\) The point \(A ( p , 4 )\) lies on \(L _ { 1 }\)
  1. Find the value of the constant \(p\). The line \(L _ { 2 }\) passes through the point \(C ( 2,4 )\) and is perpendicular to \(L _ { 1 }\)
  2. Find an equation for \(L _ { 2 }\) giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(L _ { 1 }\) and the line \(L _ { 2 }\) intersect at the point \(D\).
  3. Find the coordinates of the point \(D\).
  4. Show that the length of \(C D\) is \(\frac { 3 } { 2 } \sqrt { } 5\) A point \(B\) lies on \(L _ { 1 }\) and the length of \(A B = \sqrt { } ( 80 )\) The point \(E\) lies on \(L _ { 2 }\) such that the length of the line \(C D E = 3\) times the length of \(C D\).
  5. Find the area of the quadrilateral \(A C B E\).
Edexcel C1 2013 June Q4
5 marks Moderate -0.8
4. The line \(L _ { 1 }\) has equation \(4 x + 2 y - 3 = 0\)
  1. Find the gradient of \(L _ { 1 }\). The line \(L _ { 2 }\) is perpendicular to \(L _ { 1 }\) and passes through the point \(( 2,5 )\).
  2. Find the equation of \(L _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C1 2016 June Q10
12 marks Moderate -0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b0413ecc-b780-4f77-b76a-da7c699c12cb-12_593_1166_260_397} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The points \(P ( 0,2 )\) and \(Q ( 3,7 )\) lie on the line \(l _ { 1 }\), as shown in Figure 2.
The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\), passes through \(Q\) and crosses the \(x\)-axis at the point \(R\), as shown in Figure 2. Find
  1. an equation for \(l _ { 2 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers,
  2. the exact coordinates of \(R\),
  3. the exact area of the quadrilateral \(O R Q P\), where \(O\) is the origin.
Edexcel C1 2017 June Q8
8 marks Moderate -0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c1b0a49d-9def-4289-a4cd-288991f67caf-16_659_1438_267_251} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The straight line \(l _ { 1 }\), shown in Figure 1, has equation \(5 y = 4 x + 10\) The point \(P\) with \(x\) coordinate 5 lies on \(l _ { 1 }\) The straight line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(P\).
  1. Find an equation for \(l _ { 2 }\), writing your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers. The lines \(l _ { 1 }\) and \(l _ { 2 }\) cut the \(x\)-axis at the points \(S\) and \(T\) respectively, as shown in Figure 1.
  2. Calculate the area of triangle SPT.
Edexcel C1 Q8
10 marks Moderate -0.8
8. The straight line \(l _ { 1 }\) has equation \(y = 3 x - 6\).
The straight line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point (6, 2).
  1. Find an equation for \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
    (3)
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\).
  2. Use algebra to find the coordinates of \(C\).
    (3)
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) cross the \(x\)-axis at the points \(A\) and \(B\) respectively.
  3. Calculate the exact area of triangle \(A B C\).
    (4) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \end{tabular} & Leave blank
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    8. continuedLeave blank
    \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
Edexcel C1 2010 January Q3
5 marks Moderate -0.8
The line \(l _ { 1 }\) has equation \(3 x + 5 y - 2 = 0\)
  1. Find the gradient of \(l _ { 1 }\). The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 3,1 )\).
  2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
OCR C1 2005 June Q9
11 marks Moderate -0.3
9
  1. Find the gradient of the line \(l _ { 1 }\) which has equation \(4 x - 3 y + 5 = 0\).
  2. Find an equation of the line \(l _ { 2 }\), which passes through the point ( 1,2 ) and which is perpendicular to the line \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\). The line \(l _ { 1 }\) crosses the \(x\)-axis at \(P\) and the line \(l _ { 2 }\) crosses the \(y\)-axis at \(Q\).
  3. Find the coordinates of the mid-point of \(P Q\).
  4. Calculate the length of \(P Q\), giving your answer in the form \(\frac { \sqrt { } a } { b }\), where \(a\) and \(b\) are integers.
OCR MEI C1 Q8
3 marks Moderate -0.8
8 Find the equation of the line that passes through the point \(( 1,2 )\) and is perpendicular to the line \(3 x + 2 y = 5\).
OCR C1 Q2
4 marks Moderate -0.3
  1. The straight line \(l\) has the equation \(x - 5 y = 7\).
The straight line \(m\) is perpendicular to \(l\) and passes through the point \(( - 4,1 )\).
Find an equation for \(m\) in the form \(y = m x + c\).
OCR C1 Q7
8 marks Moderate -0.8
7. The straight line \(l\) passes through the point \(P ( - 3,6 )\) and the point \(Q ( 1 , - 4 )\).
  1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The straight line \(m\) has the equation \(2 x + k y + 7 = 0\), where \(k\) is a constant.
    Given that \(l\) and \(m\) are perpendicular,
  2. find the value of \(k\).
Edexcel C1 2014 June Q9
10 marks Moderate -0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-12_675_863_267_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The line \(l _ { 1 }\), shown in Figure 2 has equation \(2 x + 3 y = 26\) The line \(l _ { 2 }\) passes through the origin \(O\) and is perpendicular to \(l _ { 1 }\)
  1. Find an equation for the line \(l _ { 2 }\) The line \(l _ { 2 }\) intersects the line \(l _ { 1 }\) at the point \(C\).
    Line \(l _ { 1 }\) crosses the \(y\)-axis at the point \(B\) as shown in Figure 2.
  2. Find the area of triangle \(O B C\). Give your answer in the form \(\frac { a } { b }\), where \(a\) and \(b\) are integers to be determined.
OCR MEI C1 2012 January Q1
3 marks Easy -1.2
1 Find the equation of the line which is perpendicular to the line \(y = 5 x + 2\) and which passes through the point \(( 1,6 )\). Give your answer in the form \(y = a x + b\).