Equation of line through two points

1. The share price of a company is monitored.

Exactly 3 years after monitoring began, the share price was \(\pounds 1.05\) Exactly 5 years after monitoring began, the share price was \(\pounds 1.65\) The share price, \(\pounds V\), of the company is modelled by the equation $$V = p t + q$$ where \(t\) is the number of years after monitoring began and \(p\) and \(q\) are constants.

1. Find the value of \(p\) and the value of \(q\). Exactly \(T\) years after monitoring began, the share price was \(\pounds 2.50\)
2. Find the value of \(T\), according to the model, giving your answer to one decimal place.

1. A line \(l\) passes through the points \(A ( 5 , - 2 )\) and \(B ( 1,10 )\).

Find the equation of \(l\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants.
(3)

4. The point \(A ( - 6,4 )\) and the point \(B ( 8 , - 3 )\) lie on the line \(L\).

1. Find an equation for \(L\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
2. Find the distance \(A B\), giving your answer in the form \(k \sqrt { 5 }\), where \(k\) is an integer.

8.

1. Find an equation of the line joining \(A ( 7,4 )\) and \(B ( 2,0 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
2. Find the length of \(A B\), leaving your answer in surd form. The point \(C\) has coordinates ( \(2 , t\) ), where \(t > 0\), and \(A C = A B\).
3. Find the value of \(t\).
4. Find the area of triangle \(A B C\). \(\_\_\_\_\)

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.

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

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

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.

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.

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.

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.

5. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).

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 the origin and has gradient - 4 .
2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
3. Find the exact coordinates of the mid-point of \(A C\).

3. The points \(A\) and \(B\) have coordinates \(( 1,2 )\) and \(( 5,8 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\).
2. Find, in the form \(y = m x + c\), an equation for the straight line through \(A\) and \(B\).

8. 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.

12 Show that the equation of the line in Fig. C2 is \(r y + h x = h r\), as given in line 24.

3

1. The points \(A\) and \(B\) have coordinates ( \(- 4,4\) ) and ( 8,1 ) respectively. Find the equation of the line \(A B\). Give your answer in the form \(y = m x + c\).
2. Determine, with a reason, whether the line \(y = 7 - 4 x\) is perpendicular to the line \(A B\).

1 Find the equation of the line which passes through the points \(( 2,5 )\) and \(( 8 , - 1 )\). Show that this line also passes through the point \(( - 2,9 )\).

3

1. The points \(A\) and \(B\) have coordinates \(( - 4,4 )\) and \(( 8,1 )\) respectively. Find the equation of the line \(A B\). Give your answer in the form \(y = m x + c\).
2. Determine, with a reason, whether the line \(y = 7 - 4 x\) is perpendicular to the line \(A B\).

Find, in the form \(y = mx + c\), the equation of the line passing through A\((3, 7)\) and B\((5, -1)\). Show that the midpoint of AB lies on the line \(x + 2y = 10\). [5]

\includegraphics{figure_11} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).

1. Find the equation of the line through A and B. [3]
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. [2]
3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
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]

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

The line \(l\) passes through the points \(A (3, 1)\) and \(B (4, -2)\). Find an equation for \(l\). [3]

The line \(l\) passes through the points \(A(-3, 0)\) and \(B\left(\frac{5}{3}, 22\right)\)

1. Find the equation of \(l\) giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are constants. [3]

\includegraphics{figure_2} Figure 2 shows the line \(l\) and the curve \(C\), which intersect at \(A\) and \(B\). Given that

• \(C\) has equation \(y = 2x^2 + 5x - 3\)
• the region \(R\), shown shaded in Figure 2, is bounded by \(l\) and \(C\)

1. use inequalities to define \(R\). [2]