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