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

Question 9: Find the perpendicular distance from the origin of the perpendicular from the point upon the straight line .

Answer:

Given line: … … … … … i)

Therefore line perpendicular to line i) is

The perpendicular passes through . Therefore

Therefore the equation of the perpendicular line is

… … … … … ii)

Now the perpendicular distance from to line ii)

Hence the required distance is .

Question 10: Find the distance of the point from the straight line with slope and passing through the point of intersection of and .

Answer:

Given lines:

… … … … … i)

… … … … … ii)

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

The equation of line passing through and slope of :

Therefore perpendicular distance from on

Hence the required distance is

Question 11: What are the points on y-axis whose distance from the line in units?

Answer:

Let be the point on y-axis.

Given line:

Comparing with we get

Hence the required points on y-axis are and

Question 12: In the with vertices and find the equation and the length of the altitude from vertex .

Answer:

Given vertices: and

Equation of BC:

… … … … … i)

Therefore equation perpendicular to i) is

… … … … … ii)

Since this line passes thought

Therefore the equation of the perpendicular is

Perpendicular distance of from :

Hence the required distance is

Question 13: Show that the path of a moving point such that its distances from two lines and are equal is a straight line.

Answer:

Let be the point such that its distances from two lines and are equal. Therefore

Consider ve sign

Consider ve sign

Therefore the equations are and

These are also straight lines.

Question 14: If sum of perpendicular distances of a variable point from the lines and is always . Show that must move on a line.

Answer:

Let be the point such that sum of the perpendicular distance from to the given lines is . Therefore

When both are ve

This is a straight line.

Similarly, when both are ve

This is a straight line as well.

Similarly, the other two combinations are also straight lines.

Question 15: If the length of the perpendicular from the point to the line be unity, show that

Answer:

Given the perpendicular distance from point to the straight line is . Therefore,

Squaring both sides