1.03b Straight lines: parallel and perpendicular relationships

10

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 \(5 x + 3 y = 10\).
2. Points C and D have coordinates \(( - 5,4 )\) and \(( 3,6 )\) respectively. The line through C and D has equation \(4 y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB . Find the coordinates of point E .
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.

10 Fig. 10 shows a sketch of the points \(\mathrm { A } ( 2,7 ) , \mathrm { B } ( 0,3 )\) and \(\mathrm { C } ( 8 , - 1 )\). \begin{figure}[h] \end{figure}

1. Prove that angle ABC is \(90 ^ { \circ }\).
2. Find the equation of the circle which has AC as a diameter.
3. Find the equation of the tangent to this circle at A . Give your answer in the form \(a y = b x + c\), where \(a , b\) and \(c\) are integers.

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.

1. A student's attempt to solve the equation \(2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3\) is shown below.

$$\begin{aligned} & 2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3 \\ & 2 \log _ { 2 } \left( \frac { x } { \sqrt { x } } \right) = 3 \\ & 2 \log _ { 2 } ( \sqrt { x } ) = 3 \\ & \log _ { 2 } x = 3 \\ & x = 3 ^ { 2 } = 9 \end{aligned}$$ using the subtraction law for logs simplifying using the power law for logs using the definition of a log

1. Identify two errors made by this student, giving a brief explanation of each.
2. Write out the correct solution.

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}

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

15. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of a circle \(C\) with centre \(N ( 7,4 )\) The line \(l\) with equation \(y = \frac { 1 } { 3 } x\) is a tangent to \(C\) at the point \(P\).
Find

1. the equation of line \(P N\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
2. an equation for \(C\). The line with equation \(y = \frac { 1 } { 3 } x + k\), where \(k\) is a non-zero constant, is also a tangent to \(C\).
3. Find the value of \(k\).

6. \begin{figure}[h] \end{figure} The circle \(C\) has centre \(A\) with coordinates \(( - 3,1 )\).
The line \(l _ { 1 }\) with equation \(y = - 4 x + 6\), is the tangent to \(C\) at the point \(Q\), as shown in Figure 3.
a. Find the equation of the line \(A Q\) in the form \(a x + b y = c\).
b. Show that the equation of the circle \(C\) is \(( x + 3 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 17\) The line \(l _ { 2 }\) with equation \(y = - 4 x + k , k \neq 6\), is also a tangent to \(C\).
c. Find the value of the constant \(k\).

6. \begin{figure}[h] \end{figure} Not to scale The circle \(C\) has centre \(A\) with coordinates (7,5).
The line \(l\), with equation \(y = 2 x + 1\), is the tangent to \(C\) at the point \(P\), as shown in Figure 3 .

1. Show that an equation of the line \(P A\) is \(2 y + x = 17\)
2. Find an equation for \(C\). The line with equation \(y = 2 x + k , \quad k \neq 1\) is also a tangent to \(C\).
3. Find the value of the constant \(k\).

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 The vertices of triangle \(A B C\) are \(A ( - 3,1 ) , B ( 5,0 )\) and \(C ( 9,7 )\).

1. Show that \(A B = B C\).
2. Show that angle \(A B C\) is not a right angle.
3. Find the coordinates of the midpoint of \(A C\).
4. Determine the equation of the line of symmetry of the triangle, giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers to be determined.
5. Write down an equation of the circle with centre \(A\) which passes through \(B\). This circle intersects the line of symmetry of the triangle at \(B\) and at a second point.
6. Find the coordinates of this second point.

10 A triangle has vertices \(A ( 1,4 ) , B ( 7,0 )\) and \(C ( - 4 , - 1 )\).

1. Show that the equation of the line AC is \(\mathrm { y } = \mathrm { x } + 3\). M is the midpoint of AB . The line AC intersects the \(x\)-axis at D .
2. Determine the angle DMA.

8 The point A has coordinates \(( - 1 , - 2 )\) and the point B has coordinates (7,4). The perpendicular bisector of \(A B\) intersects the line \(y + 2 x = k\) at \(P\). Determine the coordinates of P in terms of \(k\).

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

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 \(C\).

3 A circle has centre \(C ( 2 , - 1 )\) and radius 5 . The point \(P\) has coordinates \(( 6,2 )\).

1. Write down an equation of the circle.
2. Verify that the point \(P\) lies on the circle.
3. Find the gradient of the line \(C P\).
4. 1. Find the gradient of a line which is perpendicular to \(C P\).
   2. Hence find an equation for the tangent to the circle at the point \(P\).

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 \(B\).