1.03b Straight lines: parallel and perpendicular relationships

1 Point A has coordinates ( 4,7 ) and point B has coordinates ( 2,1 ).

1. Find the equation of the line through A and B .
2. Point C has coordinates \(( - 1,2 )\). Show that angle \(\mathrm { ABC } = 90 ^ { \circ }\) and calculate the area of triangle ABC .
3. Find the coordinates of \(D\), the midpoint of AC. Explain also how you can tell, without having to work it out, that \(\mathrm { A } , \mathrm { B }\) and C are all the same distance from D.

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 \(y = a x + b\).

6 The points \(\mathrm { A } ( - 1,6 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { C } ( 13,4 )\) are joined by straight lines.

1. Prove that the lines AB and BC are perpendicular.
2. Find the area of triangle ABC .
3. A circle passes through the points A , B and C . Justify the statement that AC is a diameter of this circle. Find the equation of this circle.
4. Find the coordinates of the point on this circle that is furthest from B.

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

1 \begin{figure}[h] \end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).

1. Find the equation of the line through \(\mathbf { A }\) and \(\mathbf { B }\).
2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.

4 \begin{figure}[h] \end{figure} Fig. 10 shows a trapezium ABCD . The coordinates of its vertices are \(\mathrm { A } ( - 2 , - 1 ) , \mathrm { B } ( 6,3 ) , \mathrm { C } ( 3,5 )\) and \(\mathrm { D } ( - 1,3 )\).

1. Verify that the lines AB and DC are parallel.
2. Prove that the trapezium is not isosceles.
3. The diagonals of the trapezium meet at M . Find the exact coordinates of M .
4. Show that neither diagonal of the trapezium bisects the other.

5 A line has gradient - 4 and passes through the point (2,6). Find the coordinates of its points of intersection with the axes. \begin{figure}[h] \end{figure} Fig. 11 shows the line joining the points \(\mathrm { A } ( 0,3 )\) and \(\mathrm { B } ( 6,1 )\).

1. Find the equation of the line perpendicular to AB that passes through the origin, O .
2. Find the coordinates of the point where this perpendicular meets AB .
3. Show that the perpendicular distance of AB from the origin is \(\frac { 9 \sqrt { 10 } } { 10 }\).
4. Find the length of AB , expressing your answer in the form \(a \sqrt { 10 }\).
5. Find the area of triangle OAB .

1 Use coordinate geometry to answer this question. Answers obtained from accurate drawing will receive no marks. \(A\) and \(B\) are points with coordinates \(( - 1,4 )\) and \(( 7,8 )\) respectively.

1. Find the coordinates of the midpoint, M , of AB . Show also that the equation of the perpendicular bisector of AB is \(y + 2 x = 12\).
2. Find the area of the triangle bounded by the perpendicular bisector, the \(y\)-axis and the line AM , as sketched in Fig. 12.
   [diagram]

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

3 Find the coordinates of the point of intersection of the lines \(y = 3 x + 1\) and \(x + 3 y = 6\). \begin{figure}[h] \end{figure} The line AB has equation \(y = 4 x - 5\) and passes through the point \(\mathrm { B } ( 2,3 )\), 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 . \(5 \mathrm {~A} ( 9,8 ) , \mathrm { B } ( 5,0 )\) an \(\mathrm { C } ( 3,1 )\) are three points.

1. Show that AB and BC are perpendicular.
2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle.
3. BD is a diameter of the circle. Find the coordinates of D .

5

1. Find the equation of the line passing through \(\mathrm { A } ( - 1,1 )\) and \(\mathrm { B } ( 3,9 )\).
2. Show that the equation of the perpendicular bisector of AB is \(2 y + x = 11\).
3. A circle has centre \(( 5,3 )\), so that its equation is \(( x - 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = k\). Given that the circle passes through A , show that \(k = 40\). Show that the circle also passes through B .
4. Find the \(x\)-coordinates of the points where this circle crosses the \(x\)-axis. Give your answers in surd form.

7

1. Find the equation of the line passing through \(\mathrm { A } ( - 1,1 )\) and \(\mathrm { B } ( 3,9 )\).
2. Show that the equation of the perpendicular bisector of AB is \(2 y + x = 11\).
3. A circle has centre \(( 5,3 )\), so that its equation is \(( x - 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = k\). Given that the circle passes through A , show that \(k = 40\). Show that the circle also passes through B .
4. Find the \(x\)-coordinates of the points where this circle crosses the \(x\)-axis. Give your answers in surd form.

7.A circle \(C\) has centre \(X ( a , b )\) and radius \(r\) .
A line \(l\) has equation \(y = m x + c\)

1. Show that the \(x\) coordinates of the points where \(C\) and \(l\) intersect satisfy $$\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 2 ( a - m ( c - b ) ) x + a ^ { 2 } + ( c - b ) ^ { 2 } - r ^ { 2 } = 0$$ Given that \(l\) is a tangent to \(C\) ,
2. show that $$c = b - m a \pm r \sqrt { m ^ { 2 } + 1 }$$ The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 16 = 0$$ and the circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 10 y + 89 = 0$$
3. Find the equations of any lines that are normal to both \(C _ { 1 }\) and \(C _ { 2 }\) ,justifying your answer.
4. Find the equations of all lines that are a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\) [You may find the following Pythagorean triple helpful in this part: \(7 ^ { 2 } + 24 ^ { 2 } = 25 ^ { 2 }\) ]

9. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\), shown in Figure 2 has equation \(2 x + 3 y = 26\) The line \(l _ { 2 }\) passes through the origin \(O\) and is perpendicular to \(l _ { 1 }\)

1. Find an equation for the line \(l _ { 2 }\) The line \(l _ { 2 }\) 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.
2. Find the area of triangle \(O B C\). Give your answer in the form \(\frac { a } { b }\), where \(a\) and \(b\) are integers to be determined.

1 The points \(A\) and \(B\) have coordinates \(( 6,1 )\) and \(( - 2,7 )\) respectively.

1. Find the length of \(A B\).
2. Find the gradient of the line \(A B\).
3. Determine whether the line \(4 x - 3 y - 10 = 0\) is perpendicular to \(A B\).

8 The line \(l\) has gradient - 2 and passes through the point \(A ( 3,5 ) . B\) is a point on the line \(l\) such that the distance \(A B\) is \(6 \sqrt { 5 }\). Find the coordinates of each of the possible points \(B\).

9 The points \(A ( 1,3 ) , B ( 7,1 )\) and \(C ( - 3 , - 9 )\) are joined to form a triangle.

1. Show that this triangle is right-angled and state whether the right angle is at \(A , B\) or \(C\).
2. The points \(A , B\) and \(C\) lie on the circumference of a circle. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).

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

1. Find the length of \(A B\).
2. Find an equation of the line through the mid-point of \(A B\) which is perpendicular to \(A B\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

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

12 Use coordinate geometry to answer this question. Answers obtained from accurate drawing will receive no marks. \(A\) and \(B\) are points with coordinates \(( - 1,4 )\) and \(( 7,8 )\) respectively.

1. Find the coordinates of the midpoint, M , of AB . Show also that the equation of the perpendicular bisector of AB is \(y + 2 x = 12\).
2. Find the area of the triangle bounded by the perpendicular bisector, the \(y\)-axis and the line AM , as sketched in Fig. 12. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{7791371e-26d9-428c-8700-5121a1c6464a-4_451_483_776_790} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Not to scale

10 \begin{figure}[h] \end{figure} Fig. 10 shows a trapezium ABCD . The coordinates of its vertices are \(\mathrm { A } ( - 2 , - 1 ) , \mathrm { B } ( 6,3 ) , \mathrm { C } ( 3,5 )\) and \(\mathrm { D } ( - 1,3 )\).

1. Verify that the lines AB and DC are parallel.
2. Prove that the trapezium is not isosceles.
3. The diagonals of the trapezium meet at M . Find the exact coordinates of M .
4. Show that neither diagonal of the trapezium bisects the other.

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 \(y = a x + b\).

11 The points \(A ( - 1,6 ) , B ( 1,0 )\) and \(C ( 13,4 )\) are joined by straight lines.

1. Prove that the lines AB and BC are perpendicular.
2. Find the area of triangle ABC .
3. A circle passes through the points A , B and C . Justify the statement that AC is a diameter of this circle. Find the equation of this circle.
4. Find the coordinates of the point on this circle that is furthest from B .

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

10 Point A has coordinates (4, 7) and point B has coordinates (2, 1).

1. Find the equation of the line through A and B .
2. Point C has coordinates ( \(- 1,2\) ). Show that angle \(\mathrm { ABC } = 90 ^ { \circ }\) and calculate the area of triangle ABC .
3. Find the coordinates of D , the midpoint of AC . Explain also how you can tell, without having to work it out, that \(\mathrm { A } , \mathrm { B }\) and C are all the same distance from D.