Question 1: Find the distance of the point from the straight line
.
Answer:
Given equation:
Therefore perpendicular distance from point
Question 2: Find the perpendicular distance of the line joining the points from the origin.
Answer:
The equation of the line joining :
Therefore,
Question 3: Find the length of the perpendicular from the origin to the straight line joining the two points whose coordinates are .
Answer:
The equation of the line joining :
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
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
are equal to each other.
Question 5: Find the distance of the point of intersection of the lines from the line
.
Answer:
Given lines:
… … … … … i)
… … … … … ii)
Solving i) and ii) we get the point of intersection as
Comparing with
Therefore perpendicular distance from point
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
.
Answer:
Given lines:
… … … … … i)
… … … … … ii)
:
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.
Therefore perpendicular distance from point
Squaring both sides we get
Hence the required points on x-axis are
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:
… … … … … 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)
.
Question 10: Find the distance of the point from the straight line with slope
and passing through the point of intersection of
.
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
Question 11: What are the points on y-axis whose distance from the line in
units?
Answer:
Let be the point on y-axis.
Hence the required points on y-axis are
Question 12: In the with vertices
find the equation and the length of the altitude from vertex
.
Answer:
Given vertices:
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 :
Question 13: Show that the path of a moving point such that its distances from two lines are equal is a straight line.
Answer:
Let be the point such that its distances from two lines
are equal. Therefore
Consider ve sign
Consider ve sign
Therefore the equations are
These are also straight lines.
Question 14: If sum of perpendicular distances of a variable point from the lines
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