Question 1: Which of the following points lie on the line :

i) ii) iii) iv) v) vi)

Answer:

i) Substituting in we get

which is true.

Therefore point satisfies the equation and therefore lies on the equation.

ii) Substituting in we get

which is NOT true.

Therefore point does not satisfies the equation and therefore does not lies on the equation.

iii) Substituting in we get

which is true.

Therefore point satisfies the equation and therefore lies on the equation.

iv) Substituting in we get

which is true.

Therefore point satisfies the equation and therefore lies on the equation.

v) Substituting in we get

which is NOT true.

Therefore point does not satisfies the equation and therefore does not lies on the equation.

vi) Substituting in we get

which is NOT true.

Therefore point does not satisfies the equation and therefore does not lies on the equation.

Question 2: State, true or false:

i) the line passes through the point

ii) the line passes through the point

iii) the point lies on the line

iv) the point lies on the line

v) if the point lies on the line , then

Answer:

i) Substituting in we get

which is NOT true.

Therefore point satisfies the equation and therefore the equation does not pass through the given point.

ii) Substituting in we get

which is true.

Therefore point satisfies the equation and therefore the equation does not pass through the given point.

iii) Substituting in we get

which is true.

Therefore point satisfies the equation and therefore the equation passes through the given point.

iv) Substituting in we get

which is true.

Therefore point satisfies the equation and therefore the equation passes through the given point.

v) Substituting in we get

Therefore point satisfies the equation if and not when .

Question 3: The line given by the equation passes through the point ; calculate the value of .

Answer:

Substituting in we get

Therefore if point satisfies the equation then k = 4.5

Question 4: For what value of will the point lie on the line

Answer:

Substituting in we get

Therefore if point satisfies the equation then

Question 5: The line contains the point ; calculate the value of .

Answer:

Substituting in we get

Therefore if point satisfies the equation then

Question 6: Does the line bisect the join of ?

Answer:

Ratio for being a midpoint:

Let the coordinates of the point

Therefore

Therefore

Substituting in we get

which is true.

Therefore point satisfies the equation.

Hence the mid point is on the given line and bisects the given points.

Question 7:

i) The line bisects the join of , find the value of .

ii) The line bisects the join of . Find the value of .

Answer:

i) Ratio for being a midpoint:

Let the coordinates of the point

Therefore

Therefore

Substituting in we get

ii) Ratio for being a midpoint:

Let the coordinates of the point

Therefore

Therefore

Substituting in we get

Question 8:

i) The point lies on the line , calculate the value of .

ii) The line contains the point , calculate the value of .

Answer:

i) Substituting in we get

ii) Substituting in we get

Question 9: The point divides the join of in the ratio of . Does lie on the line ?

Answer:

Ratio:

Let the coordinates of the point

Therefore

Therefore

Substituting in we get

which is true.

Therefore point satisfies the equation and therefore lies on the equation.

Question 10: The line segment joining the points is divided by the point in the ratio . Does the line contain ?

Answer:

Ratio:

Let the coordinates of the point

Therefore

Therefore

Substituting in we get

which is true.

Therefore point satisfies the equation and therefore lies on the equation.

Question 11: Find the point of intersection of the lines and . If this point lies on the line , find the value of .

Answer:

Solving the two equations: and

Substituting

Therefore

Hence the point of intersection

Substituting in we get

Question 12: Show that the lines , and are concurrent.

Answer:

Solving the two equations first: , and

Substituting

Therefore

Hence the point of intersection of the two lines is

If satisfies , then the three lines are concurrent.

Substituting in we get

Hence the lines are concurrent.