1.03a Straight lines: equation forms y=mx+c, ax+by+c=0

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

1. Find the equation of the line \(A B\), giving your answer in the form \(a x + b y + c = 0\).
2. Find the coordinates of the mid-point of \(A B\). The point \(C\) has coordinates (-3,4).
3. Calculate the length of \(A C\), giving your answer in simplified surd form.
4. Determine whether the line \(A C\) is perpendicular to the line \(B C\), showing all your working.

9

1. Find the gradient of the line \(l _ { 1 }\) which has equation \(4 x - 3 y + 5 = 0\).
2. Find an equation of the line \(l _ { 2 }\), which passes through the point ( 1,2 ) and which is perpendicular to the line \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\). The line \(l _ { 1 }\) crosses the \(x\)-axis at \(P\) and the line \(l _ { 2 }\) crosses the \(y\)-axis at \(Q\).
3. Find the coordinates of the mid-point of \(P Q\).
4. Calculate the length of \(P Q\), giving your answer in the form \(\frac { \sqrt { } a } { b }\), where \(a\) and \(b\) are integers.

9 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 4 .

1. Find the centre of the circle and the value of k . The points \(\mathrm { A } ( 3 , \mathrm { a } )\) and \(\mathrm { B } ( - 1,0 )\) lie on the circumference of the circle, with \(\mathrm { a } > 0\).
2. Calculate the length of \(A B\), giving your answer in simplified surd form.
3. Find an equation for the line \(A B\).

4 Find, algebraically, the coordinates of the point of intersection of the lines \(y = 2 x - 5\) and \(6 x + 2 y = 7\).

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 Find the equation of the line passing through \(( - 1 , - 9 )\) and \(( 3,11 )\). Give your answer in the form \(y = m x + c\).

2

1. Find the points of intersection of the line \(2 x + 3 y = 12\) with the axes.
2. Find also the gradient of this line.

12

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.

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.

10 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).

1. Sketch the curve \(y = \mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.

1 Find the equation of the line which passes through \(( 1,3 )\) and ( 4,9 ).

10 \begin{figure}[h] \end{figure} In Fig.10, A has coordinates \(( 1,1 )\) and C has coordinates \(( 3,5 )\). M is the mid-point of AC . The line \(l\) is perpendicular to AC.

1. Find the coordinates of M . Hence find the equation of \(l\).
2. The point B has coordinates \(( - 2,5 )\). Show that B lies on the line \(l\).
   Find the coordinates of the point D such that ABCD is a rhombus.
3. Find the lengths MC and MB . Hence calculate the area of the rhombus ABCD .

4 The coordinates of the points \(\mathrm { A } , \mathrm { B }\) and C are ( \(- 2,2\) ), ( 1,3 ) and ( \(3 , - 3\) ) respectively.

1. Find the gradients of the lines AB and BC .
2. Show that the triangle ABC is a right-angled triangle.
3. Find the area of the triangle ABC .

8 Find the equation of the line that passes through the point \(( 1,2 )\) and is perpendicular to the line \(3 x + 2 y = 5\).

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

12 ABCD is a parallelogram. The coordinates of \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D are (-2, 3), (2, 4), (8, -3) and ( \(4 , - 4\) ) respectively. \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-4_592_725_387_492}

1. Prove that AB and BD are perpendicular.
2. Find the lengths of AB and BD and hence find the area of the parallelogram ABCD
3. Find the equation of the line CD and show that it meets the \(y\)-axis at \(\mathrm { X } ( 0 , - 5 )\).
4. Show that the lines BX and AD bisect each other.
5. Explain why the area of the parallelogram ABCD is equal to the area of the triangle BXC.
   Find the length of BX and hence calculate exactly the perpendicular distance of C from BX .

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

13 In Fig.13, XP and XQ are the perpendicular bisectors of AC and BC respectively. \begin{figure}[h] \end{figure}

1. Find the coordinates of X .
2. Hence show that \(\mathrm { AX } = \mathrm { BX } = \mathrm { CX }\).
3. The circumcircle of a triangle is the circle which passes through the vertices of the triangle.
   Write down the equation of the circumcircle of the triangle ABC .
4. Find the coordinates of the points where the circle cuts the \(x\) axis.

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 \(A B C\).

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.

1. The straight line \(l\) has the equation \(x - 5 y = 7\).

The straight line \(m\) is perpendicular to \(l\) and passes through the point \(( - 4,1 )\).
Find an equation for \(m\) in the form \(y = m x + c\).

10. \includegraphics[max width=\textwidth, alt={}, center]{af6fdbed-fcab-4db8-9cdf-fd049ce720fd-3_668_787_918_431} The diagram shows the circle \(C\) and the straight line \(l\).
The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).

1. Find the gradient of 1 .
2. Find the coordinates of the mid-point of \(A B\).
3. Find the coordinates of the centre of \(C\).
4. Show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0$$

6. The curve with equation \(y = x ^ { 2 } + 2 x\) passes through the origin, \(O\).

1. Find an equation for the normal to the curve at \(O\).
2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again.

7. A circle has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).

1. Find the length of the diameter of the circle.
2. Find an equation for the circle.
3. Show that the line \(y = 2 x - 3\) is a tangent to the circle and find the coordinates of the point of contact.

9. The straight line \(l _ { 1 }\) passes through the point \(A ( - 2,5 )\) and the point \(B ( 4,1 )\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The straight line \(l _ { 2 }\) passes through \(B\) and is perpendicular to \(l _ { 1 }\).
2. Find an equation for \(l _ { 2 }\). Given that \(l _ { 2 }\) meets the \(y\)-axis at the point \(C\),
3. show that triangle \(A B C\) is isosceles.