Question 1: Find the distance of the point from the straight line .

Answer:

Given equation:

Comparing with   we get

Therefore Perpendicular Distance from point   from

Thus the required distance is

Question 2: Find the perpendicular distance of the line joining the points and from the origin.

Answer:

The equation of the line joining and :

Comparing with   we get

Therefore,

 

 

 

 

 

 

Thus the required distance is

Question 3: Find the length of the perpendicular from the origin to the straight line joining the two points whose coordinates are and .

Answer:

The equation of the line joining and :

Comparing with   we get

Therefore the perpendicular distance from (0,0) is:

Question 4: Show that the perpendiculars let fall from any point on the straight line upon the two straight lines and are equal to each other.

Answer:

Given equations:

     … … … … … i)

     … … … … … ii)

Let be the point on

Therefore, the distance of from line i)

     … … … … … iii)

Similarly, the distance of from line ii)

     … … … … … iv)

Since is on

Substituting the value of in iii) and iv) we get

Hence the perpendicular drawn from any point on the straight line upon the two straight lines and are equal to each other.

Question 5: Find the distance of the point of intersection of the lines and from the line .

Answer:

Given lines:

     … … … … … i)

     … … … … … ii)

Solving i) and ii) we get the point of intersection as

Comparing with   we get

Therefore perpendicular distance from point   from

Thus the required distance is

Question 6: Find the length of the perpendicular from the point to the line joining the origin and the point of intersection of the line  and .

Answer:

Given lines:

     … … … … … i)

     … … … … … ii)

Solving i) and ii) we get the point of intersection as

Therefore the equation of line passing between and :

Question 7: What are the points on x-axis whose perpendicular distance from the straight line is ?

Answer:

Let the point be on the x-axis.

Given line:

Comparing with   we get

Therefore perpendicular distance from point

 

Squaring both sides we get

Hence the required points on x-axis are

and

Question 8: Show that the product of perpendiculars on the line                            from the points is .

Answer:

Let be the perpendicular distance from on

 

Let be the perpendicular distance from on