Note: The equation of a straight line passing through and making an angle
with the positive x-axis is given by
, where
is the distance of the point
on the line from the point
.
The coordinate of any point on the line at a distance from the point
are
.
Question 1: A line passes through a point and makes an angle of
with the x-axis and intersects the line
at the point
. Find
.
Answer:
Given and
The equation passing through and making an angle
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 2: If the straight line through the point makes an angle
with the x-axis and meets the line
at
, find the length
.
Answer:
Given and
The equation passing through and making an angle
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 3: A straight line drawn through the point, making an angle
with positive x-axis intersects another line
in the point
. Find length
.
Answer:
Given and
The equation passing through and making an angle
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 4: A line drawn through parallel to the line
. Find the coordinates of the two points on this line which are at a distance of
units from
.
Answer:
Given
comparing with
and
The equation passing through and and having slope of
is
Considering sign we get:
Hence the point is
Considering sign we get:
Hence the point is
Question 5: The straight line through inclined at an angle \theta with the x-axis meets the line
in
. Find the length of
.
Answer:
Given and
The equation passing through and making an angle
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 6: Find the distance of the point from the line
measured along a line making an angle of
with the x-axis.
Answer:
Given and
The equation passing through and making an angle
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 7: Find the distance of the point from the line
measured parallel to a line having slope
.
Answer:
Given
and
The equation passing through and and having slope of
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 8: Find the distance of the point from the line
measured parallel to a line having slope
.
Answer:
Given
and
The equation passing through and and having slope of
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 9: Find the distance of the point from the line
measured parallel to the line
.
Answer:
Given line:
Therefore
and
The equation passing through and and having slope of
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 10: Find the distance of the point from the line
measured parallel to the line
.
Answer:
Given
Now,
and
The equation passing through and and having slope of
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 11: Find the distance of the line from the point
in the direction of the line whose slope is
.
Answer:
Given
The equation passing through and and having slope of
is
… … … … … i)
From equation i)
The coordinates of any point on this line are
Since lies on
, substituting we get
Question 12: A line is such that its segment between the straight line and
is bisected at the point
. Obtain its equation.
Answer:
Let be the intercept between lines
and
Let makes an angle
with positive x-axis.
Given and
The equation passing through and making an angle
is
Let
Therefore the coordinate of and coordinate of
We know lies on
Also lies on
Since bisects
and
Therefore the equation of the line would be
Question 13: Find the equation of straight line passing through and having an intercept of length
between the straight lines
and
.
Answer:
Let be the intercept between lines
and
Let makes an angle
with positive x-axis.
Given and
The equation passing through and making an angle
is
Let
Therefore the coordinate of and coordinate of
We know lies on
We know lies on
Since
Squaring both sides
… … … … … ii)
From ii) either
When
Therefore the equation of line is
When
Therefore the equation of line is :