1.03a Straight lines: equation forms y=mx+c, ax+by+c=0

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Edexcel C12 2017 October Q10
9 marks Moderate -0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-24_863_929_255_511} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Diagram NOT drawn to scale The points \(A ( 7 , - 3 ) , B ( 7,20 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 4. The point \(D ( 10,5 )\) is the midpoint of \(A C\).
  1. Find the value of \(p\) and the value of \(q\). The line \(l\) passes through \(D\) and is perpendicular to \(A C\).
  2. Find an equation for \(l\), in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. Given that the line \(l\) intersects \(A B\) at \(E\),
  3. find the exact coordinates of \(E\).
Edexcel C12 2018 October Q7
8 marks Moderate -0.8
7. The point \(A\) has coordinates \(( - 1,5 )\) and the point \(B\) has coordinates \(( 4,1 )\). The line \(l\) passes through the points \(A\) and \(B\).
  1. Find the gradient of \(l\).
  2. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. The point \(M\) is the midpoint of \(A B\). The point \(C\) has coordinates \(( 5 , k )\) where \(k\) is a constant.
    Given that the distance from \(M\) to \(C\) is \(\sqrt { 13 }\)
  3. find the exact possible values of the constant \(k\).
Edexcel C1 2005 January Q8
9 marks Moderate -0.8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{bace07ee-1eb8-43d6-8229-152d1f74ab59-14_687_1196_280_388}
\end{figure} The points \(A ( 1,7 ) , B ( 20,7 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 2. The point \(D ( 8,2 )\) is the mid-point of \(A C\).
  1. Find the value of \(p\) and the value of \(q\). The line \(l\), which passes through \(D\) and is perpendicular to \(A C\), intersects \(A B\) at \(E\).
  2. Find an equation for \(l\), in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
  3. Find the exact \(x\)-coordinate of \(E\).
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 2008 January Q4
7 marks Moderate -0.8
4. The point \(A ( - 6,4 )\) and the point \(B ( 8 , - 3 )\) lie on the line \(L\).
  1. Find an equation for \(L\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
  2. Find the distance \(A B\), giving your answer in the form \(k \sqrt { 5 }\), where \(k\) is an integer.
Edexcel C1 2009 January Q10
11 marks Easy -1.2
  1. The line \(l _ { 1 }\) passes through the point \(A ( 2,5 )\) and has gradient \(- \frac { 1 } { 2 }\).
    1. Find an equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\).
    The point \(B\) has coordinates (-2, 7).
  2. Show that \(B\) lies on \(l _ { 1 }\).
  3. Find the length of \(A B\), giving your answer in the form \(k \sqrt { } 5\), where \(k\) is an integer. The point \(C\) lies on \(l _ { 1 }\) and has \(x\)-coordinate equal to \(p\).
    The length of \(A C\) is 5 units.
  4. Show that \(p\) satisfies $$p ^ { 2 } - 4 p - 16 = 0 .$$
Edexcel C1 2010 January Q9
13 marks Moderate -0.8
  1. (a) Factorise completely \(x ^ { 3 } - 4 x\) (b) Sketch the curve \(C\) with equation
$$y = x ^ { 3 } - 4 x ,$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. The point \(A\) with \(x\)-coordinate - 1 and the point \(B\) with \(x\)-coordinate 3 lie on the curve \(C\).
(c) Find an equation of the line which passes through \(A\) and \(B\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
(d) Show that the length of \(A B\) is \(k \sqrt { } 10\), where \(k\) is a constant to be found.
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 Q5
8 marks Moderate -0.8
5. The curve \(C\) has equation \(y = x ( 5 - x )\) and the line \(L\) has equation \(2 y = 5 x + 4\)
  1. Use algebra to show that \(C\) and \(L\) do not intersect.
  2. In the space on page 11, sketch \(C\) and \(L\) on the same diagram, showing the coordinates of the points at which \(C\) and \(L\) meet the axes.
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 Q10
10 marks Moderate -0.3
10. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \neq 0\), passes through the point ( \(3,7 \frac { 1 } { 2 }\) ). Given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 3 } { x ^ { 2 } }\),
  1. find \(\mathrm { f } ( x )\).
  2. Verify that \(f ( - 2 ) = 5\).
  3. Find an equation for the tangent to \(C\) at the point ( \(- 2,5\) ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
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 2010 June Q8
8 marks Moderate -0.3
8. (a) Find an equation of the line joining \(A ( 7,4 )\) and \(B ( 2,0 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
(b) Find the length of \(A B\), leaving your answer in surd form. The point \(C\) has coordinates ( \(2 , t\) ), where \(t > 0\), and \(A C = A B\).
(c) Find the value of \(t\).
(d) Find the area of triangle \(A B C\). \(\_\_\_\_\)
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 2015 June Q6
10 marks Standard +0.3
  1. The curve \(C\) has equation
$$y = \frac { \left( x ^ { 2 } + 4 \right) ( x - 3 ) } { 2 x } , \quad x \neq 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
  2. Find an equation of the tangent to \(C\) at the point where \(x = - 1\) Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
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.