Question 1: Find the equation of a straight line through the point of intersection of the lines and and parallel to .

Answer:

The equation of a straight line passing through the point of intersection of and is given by:

… … … … … i)

… … … … … ii)

Slope of line is

Since lines are parallel to each other,

Substituting the value of in i) we get the equation of the required line.

Question 2: Find the equation of a straight line passing through the point of intersection of and and perpendicular to the straight line .

Answer:

The equation of a straight line passing through the point of intersection of and is given by:

… … … … … i)

… … … … … ii)

Slope of line is

Since the two lines are perpendicular to each other,

Substituting the value of in i) we get the equation of the required line.

Question 3: Find the equation of the line passing through the point of intersection of and and is parallel to (i) x-axis (ii) y-axis

Answer:

The equation of a straight line passing through the point of intersection of and is given by:

… … … … … i)

… … … … … ii)

i) When line is parallel to x-axis, then its slope is 0. Therefore

Substituting the value of in i) we get the equation of the required line.

ii) When line is parallel to y-axis, then . Therefore

Substituting the value of in i) we get the equation of the required line.

Question 4: Find the equation of the straight line passing through the point of intersection of and and equally inclined to the axes.

Answer:

The equation of a straight line passing through the point of intersection of and is given by:

… … … … … i)

… … … … … ii)

Since the line is equally inclined to both axis, the slope of the line would be either

Case 1: Slope

Substituting the value of in i) we get the equation of the required line.

Case 2: Slope

Substituting the value of in i) we get the equation of the required line.

Question 5: Find the equation of the straight line drawn through the point of intersection of the lines and and perpendicular to the line cutting off intercepts on the axes.

Answer:

The equation of a straight line passing through the point of intersection of and is given by:

… … … … … i)

… … … … … ii)

The equation of the line with intercepts of and on axis is

… … … … … iii)

Therefore the slope iii) is

Since lines i) and iii) are perpendicular to each other, therefore

Substituting the value of in i) we get the equation of the required line.

Question 6: Prove that the family of lines represented by , being arbitrary, pass through a fixed point. Also, find the fixed point.

Answer:

Given:

This line is of the form

Therefore this line passes through the intersection of and

Solving the above two equations, we get the point of intersection as which is a fixed point through which the given family of lines passes for any value of .

Question 7: Show that the straight lines given by for different values of pass through a fixed point. Also, find that point.

Answer:

Given:

This line is of the form

Therefore this line passes through the intersection of and

Solving the above two equations, we get the point of intersection as which is a fixed point through which the given family of lines passes for any value of .

Question 8: Find the equation of the straight line passing through the point of intersection of and and making with the coordinate axes a triangle of area sq. units.

Answer:

The equation of a straight line passing through the point of intersection of and is given by:

… … … … … i)

… … … … … ii)

Therefore the point of intersection of this line with the axis are and

It is given that the required line makes an area sq. units with the coordinate axis. Therefore,

or

Substituting the value of in i) we get the equation of the required line.

Substituting the value of in i) we get the equation of the required line.

Question 9: Find the equation of the straight line which passes through the point of intersection of the lines and and makes equal and positive intercepts on the axes.

Answer:

The equation of a straight line passing through the point of intersection of and is given by:

… … … … … i)

… … … … … ii)

Since this line makes equal intercepts,

Substituting the value of in i) we get the equation of the required line.

Question 10: Find the equations of the lines through the point of intersection of the lines and and whose distance from the origin is .

Answer:

The equation of a straight line passing through the point of intersection of and is given by:

… … … … … i)

Give, distance from origin is . Therefore

Squaring both sides we get

Substituting the value of in i) we get the equation of the required line.

Question 11: Find the equations of the lines through the point of intersection of the lines and whose distance from the point is .

Answer:

The equation of a straight line passing through the point of intersection of and is given by:

… … … … … i)

Give, distance from origin is . Therefore

Squaring both sides we get

Substituting the value of in i) we get the equation of the required line.

Substituting the value of in i) we get the equation of the required line.