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.