Parallel line through point

3 Two points \(A\) and \(B\) have coordinates ( \(3 a , - a\) ) and ( \(- a , 2 a\) ) respectively, where \(a\) is a positive constant.

1. Find the equation of the line through the origin parallel to \(A B\).
2. The length of the line \(A B\) is \(3 \frac { 1 } { 3 }\) units. Find the value of \(a\).

2 The point \(M\) is the mid-point of the line joining the points \(( 3,7 )\) and \(( - 1,1 )\). Find the equation of the line through \(M\) which is parallel to the line \(\frac { x } { 3 } + \frac { y } { 2 } = 1\).

1. Given that

• the point \(A\) has coordinates \(( 4,2 )\)
• the point \(B\) has coordinates \(( 15,7 )\)
• the line \(l _ { 1 }\) passes through \(A\) and \(B\)
  1. find an equation for \(l _ { 1 }\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found.

The line \(l _ { 2 }\) passes through \(A\) and is parallel to the \(x\)-axis.
The point \(C\) lies on \(l _ { 2 }\) so that the length of \(B C\) is \(5 \sqrt { 5 }\)

Find both possible pairs of coordinates of the point \(C\).

Hence find the minimum possible area of triangle \(A B C\).

1. The line \(l _ { 1 }\) has equation

$$10 x - 2 y + 7 = 0$$

1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 1 } { 3 } , \frac { 4 } { 3 } \right)\).
2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

1. The line \(l _ { 1 }\) has equation \(2 x + 3 y = 6\)

The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(( 3 , - 5 )\).
Find the equation for the line \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

1. The line \(l _ { 1 }\) has equation

$$8 x + 2 y - 15 = 0$$

1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 3 } { 4 } , 16 \right)\).
2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

5

1. Find the gradient of the line \(4 x + 5 y = 24\).
2. A line parallel to \(4 x + 5 y = 24\) passes through the point \(( 0,12 )\). Find the coordinates of its point of intersection with the \(x\)-axis.

2 A line \(L\) is parallel to \(y = 4 x + 5\) and passes through the point \(( - 1,6 )\). Find the equation of the line \(L\) in the form \(y = a x + b\). Find also the coordinates of its intersections with the axes.

2 Find the equation of the straight line which is parallel to the line \(y = 3 x + 5\) and which goes through the point \(( 2,12 )\).

2. 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.

3 Find the equation of the line which is parallel to \(y = 5 x - 4\) and which passes through the point (2,13). Give your answer in the form \(y = a x + b\).

8 Find the equation of the line which is parallel to \(y = 3 x + 1\) and which passes through the point with coordinates \(( 4,5 )\).

2 A line has equation \(3 x + 2 y = 6\). Find the equation of the line parallel to this which passes through the point \(( 2,10 )\).

1 Find, in the form \(y = a x + b\), the equation of the line through \(( 3,10 )\) which is parallel to \(y = 2 x + 7\).

1 Find the equation of the line which is parallel to \(y = 5 x - 4\) and which passes through the point (2, 13). Give your answer in the form \(y = a x + b\).

10. \begin{figure}[h] \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 \(A B C D\).

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.

A line has equation \(3x + 2y = 6\). Find the equation of the line parallel to this which passes through the point \((2, 10)\). [3]

Find the equation of the line which is parallel to \(y = 3x + 1\) and which passes through the point with coordinates \((4, 5)\). [3]

Find, in the form \(y = ax + b\), the equation of the line through \((3, 10)\) which is parallel to \(y = 2x + 7\). [3]

The trapezium \(ABCD\) is shown below. \includegraphics{figure_2} The line \(AB\) has equation \(2x + 3y = 14\) and \(DC\) is parallel to \(AB\). The point D has coordinates \((3, 7)\).

1. Find an equation of the line DC [2 marks]
2. The angle BAD is a right angle. Find an equation of the line AD, giving your answer in the form \(mx + ny + p = 0\), where \(m\), \(n\) and \(p\) are integers. [3 marks]

\includegraphics{figure_2} The diagram shows a triangle \(ABC\). The vertices have coordinates \(A(3, -7)\), \(B(9, 1)\) and \(C(-1, -5)\).

1. 1. Find the length of the side \(AB\). [2]
   2. Find the coordinates of the mid-point of \(AB\). [1]
   3. A circle has diameter \(AB\). Find the equation of the circle in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\), \(b\) and \(r\) are constants to be found. [3]
2. Find the equation of the line \(l\) passing through \(B\) parallel to \(AC\). [3]