Question 1: Prove that the following sets of three lines are concurrent:

i) and

ii) and

iii) 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.

iii)    Given equations: and

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

and

Slope of

Therefore the slope of

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.