Question 1: Point divides the line segment joining the points in the ratio . Find the co-ordinates of point . Also, find the equation of the line through and parallel to .

Answer:

Given divides in the ratio

Ratio:

Let the coordinates of the point . Therefore

Therefore

Equation of the line given :

Therefore the required equation is

or

Question 2: The line segment joining the points is divided in the ratio at point in it. Find the co-ordinates of . Also, find the equation of the line through and perpendicular to the line .

Answer:

Given divides in the ratio

Ratio:

Let the coordinates of the point . Therefore

Therefore

Equation of the line given :

Therefore the slope of the line perpendicular to the above line

Therefore the required equation is

or

Question 3: A line meets at point . Find the co-ordinates of point . Find the equation of a line through and perpendicular to .

Answer:

At

Therefore the coordinate of

Therefore the slope of a like perpendicular to this line

Hence the line passing through with a slope of is

Question 4: Find the value of for which the lines and are perpendicular to each other. **[2003]**

Answer:

Slope of

Slope of

Since the two lines are perpendicular,

Question 5: A straight line passes through the points . It intersects the co-ordinate axes at points . is the mid-point of the line segment . Find:

- The equation of line
- The co-ordinates of
- The co-ordinates of
**[2003]**

Answer:

The equation of the line:

The and the

The coordinate of

Question 6: are the co-ordinates of vertices respectively of rhombus . Find the equations of the diagonals .

Answer:

Midpoint

Slope of

Equation of :

Slope of

Therefore equation of :

Question 7: Show that can be vertices of a square.

- Find the co-ordinates of its fourth vertex , if is a square
- Without using the co-ordinates of vertex , find the equation of side of the square and the equation of diagonal .

Answer:

Mid point of

Let the coordinate of be

In a square, the diagonals bisect each other. Therefore

Hence is

Slope of

Since , slope of

Hence the equation of :

Slope of

Hence the equation of :

Question 8: A line through origin meets the line at right angles at point . find the co-ordinates of point .

Answer:

Given … … … … (i)

Slope of line is

Slope of perpendicular

The equation of a line passing through and having slope is

… … … … (i)

Solving equations (i) and (ii)

Hence

Question 9: A straight line passes through the point and the portion of this line, intercepted between the positive axes, is bisected at this point. Find the equation of the line.

Answer:

Let y-intercept be and x-intercept be

Given is the mid point of and . Therefore:

Slope of line

Equation of line:

Question 10: Find the equation of the line passing through the point of intersection of ; and perpendicular to the line .

Answer:

Solve equations

… … … … (i)

… … … … (ii)

Multiply (i) by 4 and (ii) by 3 and then add the equations, we get

Substituting in (i) we get

Therefore the intercept is

Sloe of line is

Therefore the slope of perpendicular

Hence the equation of the perpendicular: