Note: We know that the equations of two lines passing through and making and angle with the given line are

Question 1: Find the equation of the straight lines passing through the origin and making an angle of with the straight line .

Answer:

We know that the equations of two lines passing through and making and angle with the given line are

Here

Therefore the equation are:

… … … … … i)

… … … … … ii)

Therefore are the two equations.

Question 2: Find the equations to the straight lines which pass through the origin and are inclined at an angle of to the straight line .

Answer:

Given equation:

Here

Therefore the equations are:

… … … … … i)

… … … … … ii)

Therefore are the two equations.

Question 3: Find the equations of the straight lines passing through and making an angle of with the line .

Answer:

Given equation:

Here

Therefore the equations are:

… … … … … i)

… … … … … ii)

Therefore are the two equations.

Question 4: Find the equations to the straight lines which pass through the point and are inclined at angle to the straight line .

Answer:

Here

Therefore the equations are:

… … … … … i)

… … … … … ii)

Therefore are the two equations.

Question 5: Find the equations to the straight lines passing through the point and inclined at an angle of to the line .

Answer:

Given equation:

Here

Therefore the equations are:

… … … … … i)

… … … … … ii)

Therefore are the two equations.

Question 6: Find the equations to the sides of an isosceles right angled triangle the equation of whose hypotenuse is and the opposite vertex is the point .

Answer:

Given equation:

Here

Therefore the equations are:

… … … … … i)

… … … … … ii)

Therefore are the two equations.

Question 7: The equation of one side of an equilateral triangle is and one vertex is . Prove that a second side is and find the equation of the third side.

Answer:

Refer to the adjoining figure. Since the triangle is equilateral triangle, hence all angles are

Given equation:

Here

Therefore the equations are:

… … … … … i)

… … … … … ii)

Therefore are the two equations.

Question 8: Find the equations of the two straight lines through forming two sides of a square of which is one diagonal.

Answer:

Refer to the adjoining figure.

Given equation:

Here

Therefore the equations are:

… … … … … i)

… … … … … ii)

Therefore are the two equations.

Question 9: Find the equation of two straight line passing through and making an angle of . with the line . Find also the area of the triangle formed by the three lines.

Answer:

Refer to the adjoining figure.

Given equation:

Here

Therefore the equations are:

… … … … … i)

… … … … … i)

Therefore are the two equations.

Solving &

We get

Similarly solving and

We get

Similarly, unit.

Therefore are of sq. units.

Question 10: Two sides of an isosceles triangle are given by the equations and and its third side passes through the point . Determine the equation of the third side.

Answer:

Refer to the adjoining figure.

Therefore Slope of

and Slope of

Let be the slope of

Taking the sign, we get

Also now taking ve sign,

( not possible)

Therefore the equation of sides are

… … … … … i)

… … … … … ii)

Question 11: Show that the point lies between the parallel lines and and find the equation of lines through cutting the above lines at an angle of .

Answer:

Given lines:

… … … … … i)

… … … … … ii)

The distance between line i) and ii)

Distance for from line i)

Distance for from line ii)

Now we see that

Therefore we can say that is between the two given lines.

Let m be the slope of the line passing through . This line makes with the lines. The slope of the given lines is . Therefore,

Two cases arise: Case 1:

Case 2:

Therefore, the required lines are

and

Question 12: The equation of the base of an equilateral triangle is and its vertex is . Find the length and equations of its sides.

Answer:

Since this is an equilateral triangle, hence all the three angles are .

The slope of the given lines is . Therefore,

Two cases arise: Case 1:

Case 2:

Therefore, the required lines are

and

Solving and we get

Similarly, Solving and we get

Therefore

Question 13: If two opposite vertices of a square arc and , find the coordinates of its other two vertices and the equations of its sides.

Answer:

Please refer to the adjoining figure.

Slope of

The sides and passes through and makes an angle of with whose slope is

The equation of and are given by

and

Therefore the equations of and are and respectively.

Since is parallel to , the equation of is

This line passes through . Therefore,

Therefore the equation of is

Since is parallel to , the equation of is

This line passes through . Therefore,

Therefore the equation of is

Solving equation of and we get .

Solving equation of and we get .

Hence the two vertices are and .