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 :