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

454 questions

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OCR MEI C1 2006 June Q3
3 marks Easy -1.2
A line has equation \(3x + 2y = 6\). Find the equation of the line parallel to this which passes through the point \((2, 10)\). [3]
OCR MEI C1 2009 June Q1
4 marks Moderate -0.8
A line has gradient \(-4\) and passes through the point \((2, 6)\). Find the coordinates of its points of intersection with the axes. [4]
OCR MEI C1 2009 June Q13
11 marks Moderate -0.8
A circle has equation \((x - 5)^2 + (y - 2)^2 = 20\).
  1. State the coordinates of the centre and the radius of this circle. [2]
  2. State, with a reason, whether or not this circle intersects the \(y\)-axis. [2]
  3. Find the equation of the line parallel to the line \(y = 2x\) that passes through the centre of the circle. [2]
  4. Show that the line \(y = 2x + 2\) is a tangent to the circle. State the coordinates of the point of contact. [5]
OCR MEI C1 2010 June Q1
3 marks Easy -1.2
Find the equation of the line which is parallel to \(y = 3x + 1\) and which passes through the point with coordinates \((4, 5)\). [3]
OCR MEI C1 2010 June Q11
12 marks Moderate -0.3
\includegraphics{figure_11} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).
  1. Find the equation of the line through A and B. [3]
  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. [2]
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
  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. [4]
OCR MEI C1 2011 June Q2
2 marks Easy -1.3
A line has gradient 3 and passes through the point \((1, -5)\). The point \((5, k)\) is on this line. Find the value of \(k\). [2]
OCR MEI C1 2011 June Q9
5 marks Moderate -0.3
A line \(L\) is parallel to the line \(x + 2y = 6\) and passes through the point \((10, 1)\). Find the area of the region bounded by the line \(L\) and the axes. [5]
OCR MEI C1 2012 June Q1
3 marks Easy -1.2
Find the equation of the line with gradient \(-2\) which passes through the point \((3, 1)\). Give your answer in the form \(y = ax + b\). Find also the points of intersection of this line with the axes. [3]
OCR MEI C1 2013 June Q1
3 marks Easy -1.2
Find the equation of the line which is perpendicular to the line \(y = 2x - 5\) and which passes through the point \((4, 1)\). Give your answer in the form \(y = ax + b\). [3]
Edexcel C1 Q8
10 marks Moderate -0.3
The points \(P\) and \(Q\) have coordinates \((7, 4)\) and \((9, 7)\) respectively.
  1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
The straight line \(m\) has gradient \(8\) and passes through the origin, \(O\).
  1. Write down an equation for \(m\). [1]
The lines \(l\) and \(m\) intersect at the point \(R\).
  1. Show that \(OP = OR\). [5]
Edexcel C1 Q10
13 marks Moderate -0.3
The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\). [2]
The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).
  1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]
The line \(l_2\) crosses the \(x\)-axis at the point \(C\).
  1. Find the coordinates of \(C\). [1]
The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).
  1. Show that \(D\) has coordinates \((5, 4)\). [5]
  2. Find the area of triangle \(ACD\). [2]
Edexcel C1 Q3
5 marks Moderate -0.8
The straight line \(l\) has the equation \(x - 5y = 7\). The straight line \(m\) is perpendicular to \(l\) and passes through the point \((-4, 1)\). Find an equation for \(m\) in the form \(y = mx + c\). [5]
Edexcel C1 Q8
11 marks Moderate -0.8
The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A(5, 3)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]
The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).
  1. Find the coordinates of \(B\). [3]
  2. Find the coordinates of the mid-point of \(AB\). [2]
  3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]
Edexcel C1 Q3
6 marks Moderate -0.8
The straight line \(l_1\) has the equation \(3x - y = 0\). The straight line \(l_2\) has the equation \(x + 2y - 4 = 0\).
  1. Sketch \(l_1\) and \(l_2\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes. [3]
  2. Find, as exact fractions, the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
Edexcel C1 Q10
14 marks Standard +0.3
The straight line \(l\) has gradient 3 and passes through the point \(A(-6, 4)\).
  1. Find an equation for \(l\) in the form \(y = mx + c\). [2]
The straight line \(m\) has the equation \(x - 7y + 14 = 0\). Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
  1. find the coordinates of \(B\) and \(C\), [4]
  2. show that \(\angle BAC = 90°\), [4]
  3. find the area of triangle \(ABC\). [4]
Edexcel C1 Q6
8 marks Moderate -0.8
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 \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
The straight line \(m\) has the equation \(2x + ky + 7 = 0\), where \(k\) is a constant. Given that \(l\) and \(m\) are perpendicular,
  1. find the value of \(k\). [4]
OCR C1 Q4
8 marks Moderate -0.8
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 \(y = mx + c\). [2]
The straight line \(l_2\) is perpendicular to the line with equation \(3x - y = 4\) and passes through the point with coordinates \((3, 0)\).
  1. Find an equation for \(l_2\). [3]
  2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
OCR C1 Q10
13 marks Moderate -0.3
The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).
  1. Find the coordinates of \(A\). [2]
The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).
  1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]
The line \(l_2\) crosses the \(x\)-axis at the point \(C\).
  1. Find the coordinates of \(C\). [1]
The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).
  1. Show that \(D\) has coordinates \((5, 4)\). [5]
  2. Find the area of triangle \(ACD\). [2]
OCR C1 Q9
13 marks Standard +0.8
\includegraphics{figure_9} The diagram shows the parallelogram \(ABCD\). 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 \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
The points \(C\) and \(D\) lie on the straight line \(l_2\) which has the equation \(x - 4y - 21 = 0\).
  1. Show that the distance between \(l_1\) and \(l_2\) is \(k\sqrt{17}\), where \(k\) is an integer to be found. [7]
  2. Find the area of parallelogram \(ABCD\). [2]
OCR C1 Q4
7 marks Moderate -0.3
  1. Sketch on the same diagram the graphs of \(y = (x - 1)^2(x - 5)\) and \(y = 8 - 2x\). Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
  3. State the integer, \(n\), such that $$n < \alpha < n + 1.$$ [1]
OCR C1 Q7
11 marks Moderate -0.8
The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A (5, 3)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]
The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).
  1. Find the coordinates of \(B\). [3]
  2. Find the coordinates of the mid-point of \(AB\). [2]
  3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]
OCR C1 Q5
7 marks Moderate -0.8
The straight line \(l_1\) has the equation \(3x - y = 0\). The straight line \(l_2\) has the equation \(x + 2y - 4 = 0\). \begin{enumerate}[label=(\roman*)] \item Sketch \(l_1\) and \(l_2\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes. [4] \item Find, as exact fractions, the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
OCR MEI C1 Q1
5 marks Moderate -0.8
Find the coordinates of the points of intersection of the circle \(x^2 + y^2 = 25\) and the line \(y = 3x\). Give your answers in surd form. [5]
OCR MEI C1 Q4
12 marks Moderate -0.3
\includegraphics{figure_1} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).
  1. Find the equation of the line through A and B. [3]
  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. [2]
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
  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. [4]
OCR MEI C1 Q5
14 marks Standard +0.3
  1. Points A and B have coordinates \((-2, 1)\) and \((3, 4)\) respectively. Find the equation of the perpendicular bisector of AB and show that it may be written as \(5x + 3y = 10\). [6]
  2. Points C and D have coordinates \((-5, 4)\) and \((3, 6)\) respectively. The line through C and D has equation \(4y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB. Find the coordinates of point E. [3]
  3. Find the equation of the circle with centre E which passes through A and B. Show also that CD is a diameter of this circle. [5]