Intersection of two lines

5. The line \(l _ { 1 }\) has equation \(3 y - 2 x = 30\) The line \(l _ { 2 }\) passes through the point \(A ( 24,0 )\) and is perpendicular to \(l _ { 1 }\) Lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\)

1. Find, using algebra and showing your working, the coordinates of \(P\). Given that \(l _ { 1 }\) meets the \(x\)-axis at the point \(B\),
2. find the area of triangle \(B P A\).

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}

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 \(L _ { 2 }\).

5 Find the coordinates of the point of intersection of the lines \(y = 5 x - 2\) and \(x + 3 y = 8\).

8 The lines \(y = 5 x - a\) and \(y = 2 x + 18\) meet at the point ( \(7 , b\) ).
Find the values of \(a\) and \(b\).

10. The straight line \(l\) has gradient 3 and passes through the point \(A ( - 6,4 )\).

1. Find an equation for \(l\) in the form \(y = m x + c\). The straight line \(m\) has the equation \(x - 7 y + 14 = 0\).
   Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
2. find the coordinates of \(B\) and \(C\),
3. show that \(\angle B A C = 90 ^ { \circ }\),
4. find the area of triangle \(A B C\).

1. The line \(l _ { 1 }\) has equation \(2 x + 4 y - 3 = 0\)

The line \(l _ { 2 }\) has equation \(y = m x + 7\), where \(m\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,

1. find the value of \(m\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\).
2. Find the \(x\) coordinate of \(P\). \includegraphics[max width=\textwidth, alt={}, center]{deba6a2b-1821-4110-bde8-bde18a5f9be9-02_2258_48_313_1980}

2 Show that the line which passes through the points \(( 2 , - 4 )\) and \(( - 1,5 )\) does not intersect the line \(3 x + y = 10\).

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 \(C\).

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 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 \(B\).

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 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 \(A\).

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]

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.

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}

1 The line \(A B\) has equation \(3 x + 5 y = 11\).

1. 1. Find the gradient of \(A B\).
   2. The point \(A\) has coordinates (2,1). Find an equation of the line which passes through the point \(A\) and which is perpendicular to \(A B\).
2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 8\) at the point \(C\). Find the coordinates of \(C\).

\includegraphics{figure_4} In the diagram, \(A\) is the point \((-1, 3)\) and \(B\) is the point \((3, 1)\). The line \(L_1\) passes through \(A\) and is parallel to \(OB\). The line \(L_2\) passes through \(B\) and is perpendicular to \(AB\). The lines \(L_1\) and \(L_2\) meet at \(C\). Find the coordinates of \(C\). [6]

The point \(C\) lies on the perpendicular bisector of the line joining the points \(A(4, 6)\) and \(B(10, 2)\). \(C\) also lies on the line parallel to \(AB\) through \((3, 11)\).

1. Find the equation of the perpendicular bisector of \(AB\). [4]
2. Calculate the coordinates of \(C\). [3]

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 \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]

The line \(l_2\) passes through the origin \(O\) and has gradient \(-2\). The lines \(l_1\) and \(l_2\) intersect at the point \(P\).

1. Calculate the coordinates of \(P\). [4]

Given that \(l_1\) crosses the \(y\)-axis at the point \(C\),

1. calculate the exact area of \(\triangle OCP\). [3]

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 \(y = mx + c\), where \(m\) and \(c\) are constants. [4]

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 \(S\).

1. Calculate the coordinates of \(S\). [5]
2. Show that the length of \(RS\) is \(3\sqrt{5}\). [2]
3. Hence, or otherwise, find the exact area of triangle \(PQR\). [4]