Question 1: Prove that the following sets of three lines are concurrent:
i) and
ii) and
Answer:
i) Given lines are ,
and
We have:
Hence, the given lines are concurrent.
ii) Given equations: and
We have:
Hence, the given lines are concurrent.
or the given equations are and
We have:
Hence, the given lines are concurrent.
Question 2: For what value of , are the three lines
and
concurrent?
Answer:
Given lines are and
Since the lines are concurrent,
Question 3: Find the conditions that the straight lines and
may meet in a point.
Answer:
Given lines are and
For the lines to meet at a point or for the lines to be concurrent,
Hence, the required condition is for the given lines to meet at a point.
Question 4: If the lines and
be concurrent! show that the points
and
are collinear.
Answer:
The given lines are and
If these lines are concurrent, then
Which is the condition of collinearity of three points and
Hence, if the given lines are concurrent, then the points are collinear.
Question 5: Show that the straight lines
and
are concurrent.
Answer:
Given lines are
and
If these lines are concurrent, then the determinant should be equal to
We have
Applying transformation
Applying transformation
Therefore the given lines are concurrent.
Question 6: If the three lines and
are concurrent, show that at least two of three constants
are equal.
Answer:
Given lines are and
Since the given lines are concurrent,
Applying transformation
Therefore at least two of three constants are equal
Question 7: If are in A.P., prove that the straight lines
and
are concurrent.
Answer:
Given are in A.P.
… … … … … i)
Given lines are and
For the lines to be concurrent, the determinant should be . Therefore,
Applying transformation
Substituting i) we get
Hence the lines are concurrent.
Question 8: Show that the perpendicular bisectors of the Sides of a triangle are concurrent.
Answer:
Please refer to the adjoining figure
Let the have the vertices
Let and
be the midpoints of line
and
respectively.
Therefore the coordinates of and
are
Therefore the equation of :
… … … … … i)
Similarly, the equations of BE and CF are
… … … … … ii)
… … … … … iii)
For lines i), ii) and iii) to be concurrent, the determinant should be . Therefore,
Applying transformation
Hence the three perpendicular bisectors are concurrent.