Perpendicular line through point

1 The points \(A\) and \(B\) have coordinates \(( 1,5 )\) and \(( 4,17 )\) respectively. Find the equation of the straight line which passes through the point \(( 2,8 )\) and is perpendicular to \(A B\). Give your answer in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are constants.

10. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\) has equation \(y = \frac { 3 } { 5 } x + 6\) The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(B ( 8,0 )\), as shown in the sketch in Figure 4.

1. Show that an equation for line \(l _ { 2 }\) is $$5 x + 3 y = 40$$ Given that

8. \begin{figure}[h] \end{figure} Figure 1 shows a rectangle \(A B C D\).
The point \(A\) lies on the \(y\)-axis and the points \(B\) and \(D\) lie on the \(x\)-axis as shown in Figure 1. Given that the straight line through the points \(A\) and \(B\) has equation \(5 y + 2 x = 10\)

1. show that the straight line through the points \(A\) and \(D\) has equation \(2 y - 5 x = 4\)
2. find the area of the rectangle \(A B C D\).

6 Determine the equation of the line which passes through the point \(( 4 , - 1 )\) and is perpendicular to the line with equation \(2 x + 3 y = 6\). Give your answer in the form \(y = m x + c\), where \(m\) is a fraction in its lowest terms and \(c\) is an integer.

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 \(k\).

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 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 \(B\).

2 Find the equation of the line perpendicular to the line \(y = 5 x\) which passes through the point \(( 2,11 )\). Give your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers to be found.

1 Find the equation of the line perpendicular to the line \(y = 5 x + 6\) which passes through the point \(( 1,11 )\). Give your answer in the form \(y = m x + c\).

\includegraphics{figure_2} The line \(l_1\) shown in Figure 2 has equation \(2x + 3y = 26\) The line \(l_2\) passes through the origin \(O\) and is perpendicular to \(l_1\)

1. Find an equation for the line \(l_2\) [4]

The line \(l_1\) intersects the line \(l_1\) at the point \(C\). Line \(l_1\) crosses the \(y\)-axis at the point \(B\) as shown in Figure 2.

1. Find the area of triangle \(OBC\). Give your answer in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers to be found. [6]

The line \(L\) has equation \(y = 5 - 2x\).

1. Show that the point \(P(3, -1)\) lies on \(L\). [1]
2. Find an equation of the line perpendicular to \(L\), which passes through \(P\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The points \(A\) and \(B\) have coordinates \((3, 4)\) and \((7, -6)\) respectively. The straight line \(l\) passes through \(A\) and is perpendicular to \(AB\). Find an equation for \(l\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

\includegraphics{figure_2} The straight line \(l_1\) has equation \(2y = 3x + 7\) The line \(l_1\) crosses the \(y\)-axis at the point \(A\) as shown in Figure 2.

1. 1. State the gradient of \(l_1\)
   2. Write down the coordinates of the point \(A\). [2]

Another straight line \(l_2\) intersects \(l_1\) at the point \(B(1, 5)\) and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(\angle ABC = 90°\),

1. find an equation of \(l_2\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The rectangle \(ABCD\), shown shaded in Figure 2, has vertices at the points \(A\), \(B\), \(C\) and \(D\).

1. Find the exact area of rectangle \(ABCD\). [5]

\(A\) is the point \((5, 7)\) and \(B\) is the point \((-1, -5)\).

1. Find the coordinates of the mid-point of the line segment \(AB\). [2]
2. Find an equation of the line through \(A\) that is perpendicular to the line segment \(AB\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [5]

\includegraphics{figure_7} The line AB has equation \(y = 4x - 5\) and passes through the point B(2, 3), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C. Find the equation of the line BC and the \(x\)-coordinate of C. [5]

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]

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]

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]

Find the equation of the line which is perpendicular to the line \(y = 5x + 2\) and which passes through the point \((1, 6)\). Give your answer in the form \(y = ax + b\). [3]

The line \(L\) has equation \(2x + 3y = 7\) Which one of the following is perpendicular to \(L\)? Tick one box. [1 mark] \(2x - 3y = 7\) \(3x + 2y = -7\) \(2x + 3y = -\frac{1}{7}\) \(3x - 2y = 7\)

\includegraphics{figure_2} Figure 2 is a sketch showing the line \(l_1\) with equation \(y = 2x - 1\) and the point \(A\) with coordinates \((-2, 3)\). The line \(l_2\) passes through \(A\) and is perpendicular to \(l_1\)

1. Find the equation of \(l_2\) writing your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants to be found. [3]

The point \(B\) and the point \(C\) lie on \(l_1\) such that \(ABC\) is an isosceles triangle with \(AB = AC = 2\sqrt{13}\)

1. Show that the \(x\) coordinates of points \(B\) and \(C\) satisfy the equation $$5x^2 - 12x - 32 = 0$$ [4]

Given that \(B\) lies in the 3rd quadrant

1. find, using algebra and showing your working, the coordinates of \(B\). [4]

Determine the equation of the line that passes through the point \((1, 3)\) and is perpendicular to the line with equation \(3x + 6y - 5 = 0\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]