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

454 questions

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Edexcel C1 Q6
6 marks Moderate -0.5
6. The straight line \(l\) has the equation \(x - 2 y = 12\) and meets the coordinate axes at the points \(A\) and \(B\). Find the distance of the mid-point of \(A B\) from the origin, giving your answer in the form \(k \sqrt { 5 }\).
Edexcel C1 Q7
8 marks Moderate -0.8
7. The straight line \(l _ { 1 }\) has gradient 2 and passes through the point with coordinates \(( 4 , - 5 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(y = m x + c\). The straight line \(l _ { 2 }\) is perpendicular to the line with equation \(3 x - y = 4\) and passes through the point with coordinates \(( 3,0 )\).
  2. Find an equation for \(l _ { 2 }\).
  3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
Edexcel C1 Q8
11 marks Standard +0.3
8. The straight line \(l _ { 1 }\) passes through the point \(A ( - 2,5 )\) and the point \(B ( 4,1 )\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The straight line \(l _ { 2 }\) passes through \(B\) and is perpendicular to \(l _ { 1 }\).
  2. Find an equation for \(l _ { 2 }\). Given that \(l _ { 2 }\) meets the \(y\)-axis at the point \(C\),
  3. show that triangle \(A B C\) is isosceles.
Edexcel C1 Q9
13 marks Standard +0.3
9. A curve has the equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 7 x\).
  1. Show that the curve only crosses the \(x\)-axis at one point. The point \(P\) on the curve has coordinates \(( 3,3 )\).
  2. Find an equation for the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The normal to the curve at \(P\) meets the coordinate axes at \(Q\) and \(R\).
  3. Show that triangle \(O Q R\), where \(O\) is the origin, has area \(28 \frac { 1 } { 8 }\).
Edexcel C1 Q10
13 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5e8f154b-c232-49ee-a798-f61ff08ca0b9-4_663_1113_950_402} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the parallelogram \(A B C D\).
The points \(A\) and \(B\) have coordinates \(( - 1,3 )\) and \(( 3,4 )\) respectively and lie on the straight line \(l _ { 1 }\).
  1. Find an equation for \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The points \(C\) and \(D\) lie on the straight line \(l _ { 2 }\) which has the equation \(x - 4 y - 21 = 0\).
  2. Show that the distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(k \sqrt { 17 }\), where \(k\) is an integer to be found.
  3. Find the area of parallelogram \(A B C D\).
Edexcel C1 Q1
4 marks Easy -1.2
  1. The points \(A , B\) and \(C\) have coordinates \(( - 3,0 ) , ( 5 , - 2 )\) and \(( 4,1 )\) respectively.
Find an equation for the straight line which passes through \(C\) and is parallel to \(A B\).
Give your answer in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers.
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 \(x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0\).
AQA FP1 2009 January Q7
10 marks Standard +0.3
7 The points \(P ( a , c )\) and \(Q ( b , d )\) lie on the curve with equation \(y = \mathrm { f } ( x )\). The straight line \(P Q\) intersects the \(x\)-axis at the point \(R ( r , 0 )\). The curve \(y = \mathrm { f } ( x )\) intersects the \(x\)-axis at the point \(S ( \beta , 0 )\). \includegraphics[max width=\textwidth, alt={}, center]{38c2a2c8-84cc-4bd2-b3ad-f9dee59763ba-4_951_971_470_539}
  1. Show that $$r = a + c \left( \frac { b - a } { c - d } \right)$$
  2. Given that $$a = 2 , b = 3 \text { and } \mathrm { f } ( x ) = 20 x - x ^ { 4 }$$
    1. find the value of \(r\);
    2. show that \(\beta - r \approx 0.18\).
AQA FP1 2008 June Q9
14 marks Standard +0.8
9 The diagram shows the parabola \(y ^ { 2 } = 4 x\) and the point \(A\) with coordinates \(( 3,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-05_732_657_370_689}
  1. Find an equation of the straight line having gradient \(m\) and passing through the point \(A ( 3,4 )\).
  2. Show that, if this straight line intersects the parabola, then the \(y\)-coordinates of the points of intersection satisfy the equation $$m y ^ { 2 } - 4 y + ( 16 - 12 m ) = 0$$
  3. By considering the discriminant of the equation in part (b), find the equations of the two tangents to the parabola which pass through \(A\).
    (No credit will be given for solutions based on differentiation.)
  4. Find the coordinates of the points at which these tangents touch the parabola.
CAIE P1 2016 November Q4
6 marks Easy -1.2
  1. Find the equation of the line \(C D\), giving your answer in the form \(y = m x + c\).
  2. Find the distance \(A D\).
OCR C1 2007 January Q9
12 marks Moderate -0.3
  1. Find the equation of the line through \(A\) parallel to the line \(y = 4 x - 5\), giving your answer in the form \(y = m x + c\).
  2. Calculate the length of \(A B\), giving your answer in simplified surd form.
  3. Find the equation of the line which passes through the mid-point of \(A B\) and which is perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2009 June Q9
8 marks Moderate -0.8
  1. Calculate the length of \(A B\).
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the equation of the line through \(( 1,3 )\) which is parallel to \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 Q9
10 marks Standard +0.3
  1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The straight line \(m\) has gradient 8 and passes through the origin, \(O\).
  2. Write down an equation for \(m\). The lines \(l\) and \(m\) intersect at the point \(R\).
  3. Show that \(O P = O R\).
Edexcel AEA 2006 June Q6
15 marks Challenging +1.2
  1. Show that the point \(P ( 1,0 )\) lies on \(C\) .
  2. Find the coordinates of the point \(Q\) .
  3. Find the area of the shaded region between \(C\) and the line \(P Q\) .
OCR MEI AS Paper 2 2021 November Q10
6 marks Standard +0.3
  1. Show that PQ is perpendicular to QR . A circle passes through \(\mathrm { P } , \mathrm { Q }\) and R .
  2. Determine the coordinates of the centre of the circle.
OCR H240/02 2021 November Q5
8 marks Standard +0.3
5 In this question you must show detailed reasoning. Points \(A , B\) and \(C\) have coordinates \(( 0,6 ) , ( 7,5 )\) and \(( 6 , - 2 )\) respectively.
  1. Find an equation of the perpendicular bisector of \(A B\).
  2. Hence, or otherwise, find an equation of the circle that passes through points \(A , B\) and \(C\).
OCR PURE Q6
6 marks Moderate -0.3
6 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}]{7fc02f90-8f8b-4153-bba1-dc0807124e96-4_650_661_1765_242}
The diagram shows the line \(3 y + x = 7\) which is a tangent to a circle with centre \(( 3 , - 2 )\).
Find an equation for the circle.
OCR MEI AS Paper 2 2020 November Q7
8 marks Moderate -0.3
7 In this question you must show detailed reasoning. A circle has centre \(( 2 , - 1 )\) and radius 5. A straight line passes through the points \(( 1,1 )\) and \(( 9,5 )\).
Find the coordinates of the points of intersection of the line and the circle.
OCR H240/01 2018 September Q7
11 marks Moderate -0.3
7 A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).
  1. Show that the line does not meet the circle.
  2. (a) Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
    (b) Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.
OCR H240/01 2018 December Q1
4 marks Easy -1.2
1 In this question you must show detailed reasoning. Andrea is comparing the prices charged by two different taxi firms.
Firm A charges \(\pounds 20\) for a 5 mile journey and \(\pounds 30\) for a 10 mile journey, and there is a linear relationship between the price and the length of the journey.
Firm B charges a pick-up fee of \(\pounds 3\) and then \(\pounds 2.40\) for each mile travelled.
Find the length of journey for which both firms would charge the same amount.
OCR AS Pure 2017 Specimen Q2
5 marks Moderate -0.8
2 Points \(A\) and \(B\) have coordinates \(( 3,0 )\) and \(( 9,8 )\) respectively. The line \(A B\) is a diameter of a circle.
  1. Find the coordinates of the centre of the circle.
  2. Find the equation of the tangent to the circle at the point \(B\).
Edexcel C1 Q8
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]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_59_2568_1882} \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_1829_2648_114}
Edexcel M2 Q3
11 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-02_650_1043_367_317} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The points \(A ( 3,0 )\) and \(B ( 0,4 )\) are two vertices of the rectangle \(A B C D\), as shown in Fig. 2.
  1. Write down the gradient of \(A B\) and hence the gradient of \(B C\). The point \(C\) has coordinates \(( 8 , k )\), where \(k\) is a positive constant.
  2. Find the length of \(B C\) in terms of \(k\). Given that the length of \(B C\) is 10 and using your answer to part (b),
  3. find the value of \(k\),
  4. find the coordinates of \(D\).
Edexcel M2 Q16
13 marks Moderate -0.8
16. \section*{Figure 3}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-08_581_575_395_609}
The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 3.
  1. Find the coordinates of the centre of the circle.
  2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
  4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.
AQA C1 2007 January Q2
11 marks Moderate -0.3
2 The line \(A B\) has equation \(3 x + 5 y = 8\) and the point \(A\) has coordinates (6, -2).
    1. Find the gradient of \(A B\).
    2. Hence find an equation of the straight line which is perpendicular to \(A B\) and which passes through \(A\).
  2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 3\) at the point \(B\). Find the coordinates of \(B\).
  3. The point \(C\) has coordinates \(( 2 , k )\) and the distance from \(A\) to \(C\) is 5 . Find the two possible values of the constant \(k\).