1.03b Straight lines: parallel and perpendicular relationships

322 questions

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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 2014 January Q6
11 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6081d81b-51d2-4140-9834-71ef7fd700b0-12_650_885_255_603} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line \(l _ { 1 }\) has equation \(2 y = 3 x + 7\) The line \(l _ { 1 }\) crosses the \(y\)-axis at the point \(A\) as shown in Figure 2.
    1. State the gradient of \(l _ { 1 }\)
    2. Write down the coordinates of the point \(A\). Another straight line \(l _ { 2 }\) intersects \(l _ { 1 }\) at the point \(B ( 1,5 )\) and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(\angle A B C = 90 ^ { \circ }\),
  2. find an equation of \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. The rectangle \(A B C D\), shown shaded in Figure 2, has vertices at the points \(A , B , C\) and \(D\).
  3. Find the exact area of rectangle \(A B C D\).
Edexcel C1 2005 June Q8
10 marks Moderate -0.8
8. The line \(l _ { 1 }\) passes through the point \(( 9 , - 4 )\) and has gradient \(\frac { 1 } { 3 }\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(l _ { 2 }\) passes through the origin \(O\) and has gradient - 2 . The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).
  2. Calculate the coordinates of \(P\). Given that \(l _ { 1 }\) crosses the \(y\)-axis at the point \(C\),
  3. calculate the exact area of \(\triangle O C P\). \includegraphics[max width=\textwidth, alt={}, center]{5a195cf1-37d9-43e9-ab47-c6892a18ba80-10_187_62_2563_1881}
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 2007 June Q11
9 marks Moderate -0.8
  1. The line \(l _ { 1 }\) has equation \(y = 3 x + 2\) and the line \(l _ { 2 }\) has equation \(3 x + 2 y - 8 = 0\).
    1. Find the gradient of the line \(l _ { 2 }\).
    The point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\) is \(P\).
  2. Find the coordinates of \(P\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) cross the line \(y = 1\) at the points \(A\) and \(B\) respectively.
  3. Find the area of triangle \(A B P\).
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 2013 June Q6
7 marks Moderate -0.8
6. The straight line \(L _ { 1 }\) passes through the points \(( - 1,3 )\) and \(( 11,12 )\).
  1. Find an equation for \(L _ { 1 }\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The line \(L _ { 2 }\) has equation \(3 y + 4 x - 30 = 0\).
  2. Find the coordinates of the point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
Edexcel C1 2014 June Q7
9 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6db8acbd-7f61-46ff-8fdc-f0f4a8363aa6-10_869_1073_267_440} \captionsetup{labelformat=empty} \caption{Diagram NOT to scale}
\end{figure} Figure 2 Figure 2 shows a right angled triangle \(L M N\). The points \(L\) and \(M\) have coordinates ( \(- 1,2\) ) and ( \(7 , - 4\) ) respectively.
  1. Find an equation for the straight line passing through the points \(L\) and \(M\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. Given that the coordinates of point \(N\) are ( \(16 , p\) ), where \(p\) is a constant, and angle \(L M N = 90 ^ { \circ }\),
  2. find the value of \(p\). Given that there is a point \(K\) such that the points \(L , M , N\), and \(K\) form a rectangle,
  3. find the \(y\) coordinate of \(K\).
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 2018 June Q8
8 marks Moderate -0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{937246f9-2b6a-48df-b919-c6db3d6f863b-20_1063_1319_251_365} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the straight line \(l _ { 1 }\) with equation \(4 y = 5 x + 12\)
  1. State the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the point \(E ( 12,5 )\), as shown in Figure 2.
  2. Find the equation of \(l _ { 2 }\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined. The line \(l _ { 2 }\) cuts the \(x\)-axis at the point \(C\) and the \(y\)-axis at the point \(B\).
  3. Find the coordinates of
    1. the point \(B\),
    2. the point \(C\). The line \(l _ { 1 }\) cuts the \(y\)-axis at the point \(A\).
      The point \(D\) lies on \(l _ { 1 }\) such that \(A B C D\) is a parallelogram, as shown in Figure 2.
  4. Find the area of \(A B C D\).
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
    \hline \end{tabular} \end{center}
    8. continuedLeave blank
    \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}
Edexcel P2 2019 October Q6
7 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bfeb1724-9a00-4a36-9606-520395792b2b-16_677_826_258_559} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a circle \(C\) with centre \(N ( 4 , - 1 )\). The line \(l\) with equation \(y = \frac { 1 } { 2 } x\) is a tangent to \(C\) at the point \(P\). Find
  1. the equation of line \(P N\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
  2. the equation of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{bfeb1724-9a00-4a36-9606-520395792b2b-16_2256_52_311_1978}
Edexcel P2 2023 October Q7
8 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66abdef1-072e-41eb-a933-dd51a96330ff-16_949_940_246_566} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of
  • the circle \(C\) with centre \(X ( 4 , - 3 )\)
  • the line \(l\) with equation \(y = \frac { 5 } { 2 } x - \frac { 55 } { 2 }\)
Given that \(l\) is the tangent to \(C\) at the point \(N\),
  1. show that an equation for the straight line passing through \(X\) and \(N\) is $$2 x + 5 y + 7 = 0$$
  2. Hence find
    1. the coordinates of \(N\),
    2. an equation for \(C\).
Edexcel C2 2006 June Q7
8 marks Moderate -0.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{29c7baa1-6929-448a-a756-319ea75dffa7-08_611_682_296_641}
\end{figure} The line \(y = 3 x - 4\) is a tangent to the circle \(C\), touching \(C\) at the point \(P ( 2,2 )\), as shown in Figure 1. The point \(Q\) is the centre of \(C\).
  1. Find an equation of the straight line through \(P\) and \(Q\). Given that \(Q\) lies on the line \(y = 1\),
  2. show that the \(x\)-coordinate of \(Q\) is 5,
  3. find an equation for \(C\).
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.
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}
Edexcel C2 2016 June Q3
8 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{582cda45-80fc-43a8-90e6-1cae08cb1534-05_791_917_121_484} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale The circle \(C\) has centre \(P ( 7,8 )\) and passes through the point \(Q ( 10,13 )\), as shown in Figure 2.
  1. Find the length \(P Q\), giving your answer as an exact value.
  2. Hence write down an equation for \(C\). The line \(l\) is a tangent to \(C\) at the point \(Q\), as shown in Figure 2.
  3. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel F1 2015 June Q6
10 marks Challenging +1.2
6. The rectangular hyperbola, \(H\), has cartesian equation $$x y = 36$$ The three points \(P \left( 6 p , \frac { 6 } { p } \right) , Q \left( 6 q , \frac { 6 } { q } \right)\) and \(R \left( 6 r , \frac { 6 } { r } \right)\), where \(p , q\) and \(r\) are distinct, non-zero values, lie on the hyperbola \(H\).
  1. Show that an equation of the line \(P Q\) is $$p q y + x = 6 ( p + q )$$ Given that \(P R\) is perpendicular to \(Q R\),
  2. show that the normal to the curve \(H\) at the point \(R\) is parallel to the line \(P Q\).