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

1 The point \(A\) has coordinates \(( 11,2 )\) and the point \(B\) has coordinates \(( - 1 , - 1 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence, or otherwise, show that the line \(A B\) has equation $$x - 4 y = 3$$
2. The line with equation \(3 x + 5 y = 26\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).

2 The point \(A\) has coordinates \(( 1,1 )\) and the point \(B\) has coordinates \(( 5 , k )\). The line \(A B\) has equation \(3 x + 4 y = 7\).

1. 1. Show that \(k = - 2\).
   2. Hence find the coordinates of the mid-point of \(A B\).
2. Find the gradient of \(A B\).
3. The line \(A C\) is perpendicular to the line \(A B\).
   1. Find the gradient of \(A C\).
   2. Hence find an equation of the line \(A C\).
   3. Given that the point \(C\) lies on the \(x\)-axis, find its \(x\)-coordinate.

1 The points \(A\) and \(B\) have coordinates \(( 1,6 )\) and \(( 5 , - 2 )\) respectively. The mid-point of \(A B\) is \(M\).

1. Find the coordinates of \(M\).
2. Find the gradient of \(A B\), giving your answer in its simplest form.
3. A straight line passes through \(M\) and is perpendicular to \(A B\).
   1. Show that this line has equation \(x - 2 y + 1 = 0\).
   2. Given that this line passes through the point \(( k , k + 5 )\), find the value of the constant \(k\).

2 The triangle \(A B C\) has vertices \(A ( 1,3 ) , B ( 3,7 )\) and \(C ( - 1,9 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence show that angle \(A B C\) is a right angle.
2. 1. Find the coordinates of \(M\), the mid-point of \(A C\).
   2. Show that the lengths of \(A B\) and \(B C\) are equal.
   3. Hence find an equation of the line of symmetry of the triangle \(A B C\).

3 The line \(A B\) has equation \(3 x + 2 y = 7\). The point \(C\) has coordinates \(( 2 , - 7 )\).

1. 1. Find the gradient of \(A B\).
   2. The line which passes through \(C\) and which is parallel to \(A B\) crosses the \(y\)-axis at the point \(D\). Find the \(y\)-coordinate of \(D\).
2. The line with equation \(y = 1 - 4 x\) intersects the line \(A B\) at the point \(A\). Find the coordinates of \(A\).
3. The point \(E\) has coordinates \(( 5 , k )\). Given that \(C E\) has length 5 , find the two possible values of the constant \(k\).

1 The point \(A\) has coordinates (6, -4) and the point \(B\) has coordinates (-2, 7).

1. Given that the point \(O\) has coordinates \(( 0,0 )\), show that the length of \(O A\) is less than the length of \(O B\).
2. 1. Find the gradient of \(A B\).
   2. Find an equation of the line \(A B\) in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
3. The point \(C\) has coordinates \(( k , 0 )\). The line \(A C\) is perpendicular to the line \(A B\). Find the value of the constant \(k\).

1 The point \(A\) has coordinates \(( - 3,2 )\) and the point \(B\) has coordinates \(( 7 , k )\).
The line \(A B\) has equation \(3 x + 5 y = 1\).

1. 1. Show that \(k = - 4\).
   2. Hence find the coordinates of the midpoint of \(A B\).
2. Find the gradient of \(A B\).
3. A line which passes through the point \(A\) is perpendicular to the line \(A B\). Find an equation of this line, giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
4. The line \(A B\), with equation \(3 x + 5 y = 1\), intersects the line \(5 x + 8 y = 4\) at the point \(C\). Find the coordinates of \(C\).

1 The point \(A\) has coordinates \(( 6,5 )\) and the point \(B\) has coordinates \(( 2 , - 1 )\).

1. Find the coordinates of the midpoint of \(A B\).
2. Show that \(A B\) has length \(k \sqrt { 13 }\), where \(k\) is an integer.
3. 1. Find the gradient of the line \(A B\).
   2. Hence, or otherwise, show that the line \(A B\) has equation \(3 x - 2 y = 8\).
4. The line \(A B\) intersects the line with equation \(2 x + y = 10\) at the point \(C\). Find the coordinates of \(C\).

1 The point \(A\) has coordinates \(( 1,7 )\) and the point \(B\) has coordinates \(( 5,1 )\).

1. 1. Find the gradient of the line \(A B\).
   2. Hence, or otherwise, show that the line \(A B\) has equation \(3 x + 2 y = 17\).
2. The line \(A B\) intersects the line with equation \(x - 4 y = 8\) at the point \(C\). Find the coordinates of \(C\).
3. Find an equation of the line through \(A\) which is perpendicular to \(A B\).

1 The trapezium \(A B C D\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-2_298_591_557_737} The line \(A B\) has equation \(2 x + 3 y = 14\) and \(D C\) is parallel to \(A B\).

1. Find the gradient of \(A B\).
2. The point \(D\) has coordinates \(( 3,7 )\).
   1. Find an equation of the line \(D C\).
   2. The angle \(B A D\) is a right angle. Find an equation of the line \(A D\), giving your answer in the form \(m x + n y + p = 0\), where \(m , n\) and \(p\) are integers.
3. The line \(B C\) has equation \(5 y - x = 6\). Find the coordinates of \(B\).

1 The line \(A B\) has equation \(7 x + 3 y = 13\).

1. Find the gradient of \(A B\).
2. The point \(C\) has coordinates \(( - 1,3 )\).
   1. Find an equation of the line which passes through the point \(C\) and which is parallel to \(A B\).
   2. The point \(\left( 1 \frac { 1 } { 2 } , - 1 \right)\) is the mid-point of \(A C\). Find the coordinates of the point \(A\).
3. The line \(A B\) intersects the line with equation \(3 x + 2 y = 12\) at the point \(B\). Find the coordinates of \(B\).

6 The curve with equation \(y = x ^ { 3 } - 2 x ^ { 2 } + 3\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{c44c229e-44b2-4799-9c9c-bfccdd09d450-4_590_787_365_625} The curve cuts the \(x\)-axis at the point \(A ( - 1,0 )\) and passes through the point \(B ( 1,2 )\).

1. Find \(\int _ { - 1 } ^ { 1 } \left( x ^ { 3 } - 2 x ^ { 2 } + 3 \right) \mathrm { d } x\).
   (5 marks)
2. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 2 x ^ { 2 } + 3\) and the line \(A B\).
   (3 marks)

2 The line \(A B\) has equation \(4 x - 3 y = 7\).

1. 1. Find the gradient of \(A B\).
   2. Find an equation of the straight line that is parallel to \(A B\) and which passes through the point \(C ( 3 , - 5 )\), giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
2. The line \(A B\) intersects the line with equation \(3 x - 2 y = 4\) at the point \(D\). Find the coordinates of \(D\).
3. The point \(E\) with coordinates \(( k - 2,2 k - 3 )\) lies on the line \(A B\). Find the value of the constant \(k\).

1 The line \(A B\) has equation \(3 x - 4 y + 5 = 0\).

1. The point with coordinates \(( p , p + 2 )\) lies on the line \(A B\). Find the value of the constant \(p\).
2. Find the gradient of \(A B\).
3. The point \(A\) has coordinates ( 1,2 ). The point \(C ( - 5 , k )\) is such that \(A C\) is perpendicular to \(A B\). Find the value of \(k\).
4. The line \(A B\) intersects the line with equation \(2 x - 5 y = 6\) at the point \(D\). Find the coordinates of \(D\).

1 The point \(A\) has coordinates \(( - 1,2 )\) and the point \(B\) has coordinates \(( 3 , - 5 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence find an equation of the line \(A B\), giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
2. The midpoint of \(A B\) is \(M\).
   1. Find the coordinates of \(M\).
   2. Find an equation of the line which passes through \(M\) and which is perpendicular to \(A B\). [3 marks]
3. The point \(C\) has coordinates \(( k , 2 k + 3 )\). Given that the distance from \(A\) to \(C\) is \(\sqrt { 13 }\), find the two possible values of the constant \(k\).
   [0pt] [4 marks]

1 The line \(A B\) has equation \(3 x + 5 y = 7\).

1. Find the gradient of \(A B\).
2. Find an equation of the line that is perpendicular to the line \(A B\) and which passes through the point \(( - 2 , - 3 )\). Express your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
3. The line \(A C\) has equation \(2 x - 3 y = 30\). Find the coordinates of \(A\).

2 The point \(P\) has coordinates \(( \sqrt { 3 } , 2 \sqrt { 3 } )\) and the point \(Q\) has coordinates \(( \sqrt { 5 } , 4 \sqrt { 5 } )\). Show that the gradient of \(P Q\) can be expressed as \(n + \sqrt { 15 }\), stating the value of the integer \(n\).
[0pt] [5 marks]

1 The line \(A B\) has equation \(5 x + 3 y + 3 = 0\).

1. The line \(A B\) is parallel to the line with equation \(y = m x + 7\). Find the value of \(m\).
2. The line \(A B\) intersects the line with equation \(3 x - 2 y + 17 = 0\) at the point \(B\). Find the coordinates of \(B\).
3. The point with coordinates \(( 2 k + 3,4 - 3 k )\) lies on the line \(A B\). Find the value of \(k\).
   [0pt] [2 marks]

5. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).

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 the origin and has gradient - 4 .
2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
3. Find the exact coordinates of the mid-point of \(A C\).

6. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$

1. Write down the maximum value of \(\mathrm { f } ( x )\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
5. Find the value of \(k\).

3. The points \(A\) and \(B\) have coordinates \(( 1,2 )\) and \(( 5,8 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\).
2. Find, in the form \(y = m x + c\), an equation for the straight line through \(A\) and \(B\).

9. The points \(A ( - 1 , - 2 ) , B ( 7,2 )\) and \(C ( k , 4 )\), where \(k\) is a constant, are the vertices of \(\triangle A B C\). Angle \(A B C\) is a right angle.

1. Find the gradient of \(A B\).
2. Calculate the value of \(k\).
3. Show that the length of \(A B\) may be written in the form \(p \sqrt { 5 }\), where \(p\) is an integer to be found.
4. Find the exact value of the area of \(\triangle A B C\).
5. Find an equation for the straight line \(l\) passing through \(B\) and \(C\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

8. The points \(A ( - 1 , - 2 ) , B ( 7,2 )\) and \(C ( k , 4 )\), where \(k\) is a constant, are the vertices of \(\triangle A B C\). Angle \(A B C\) is a right angle.

1. Find the gradient of \(A B\).
2. Calculate the value of \(k\).
3. Show that the length of \(A B\) may be written in the form \(p \sqrt { 5 }\), where \(p\) is an integer to be found.
4. Find the exact value of the area of \(\triangle A B C\).
5. Find an equation for the straight line \(l\) passing through \(B\) and \(C\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7. \begin{figure}[h] \end{figure} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 1.

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.

8. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$

1. Write down the maximum value of \(\mathrm { f } ( x )\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
5. Find the value of \(k\).