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:

Question 11: Find the equation of the line which is perpendicular to the line at the point where this line meets .

Answer:

Slope of line is

Therefore slope of line perpendicular to given line

Therefore the equation of line passing through (0,b) and having slope of is:

Question 12: are the vertices of a triangle . Find:

(i) The equation of the median of triangle through vertex

(ii) The equation of altitude of triangle through vertex

Answer:

Mid point of

Therefore the equation of median of through is

(i)

(ii) Slope of

Slope of line perpendicular to

Therefore the equation of altitude of through

Question 13: Determine whether the line through points is perpendicular to the line . Does line bisect the line segment joining the two given points?

Answer:

Slope of line passing through and

Slope of is

Slope of perpendicular

Therefore line passing through and is perpendicular to

Mid point of and

Substituting in we get that it satisfies the equation. Therefore bisects the line joining and

Question 14: Given a straight line . Determine the equation of the other line which is parallel to its and passes through .

Answer:

Given

Slope of this line

Equation of line with slope and passing through is

Question 15: Find the value of such that the line is:

(i) Perpendicular to the line (ii) Parallel to it.

Answer:

Given

Slope of this line

Slope of line is

Slope of line perpendicular to this line

(i) If perpendicular

(ii) If parallel

Question 16: The vertices of a triangle are . Write down the equation of . Find:

(i) The equation of the line through and perpendicular to .

(ii) The co-ordinates of the point , where the perpendicular through , as obtained in (i.), meets .

Answer:

(i) Slope of

Slope of line perpendicualr to

Therefore equation of line passing through with slope is:

… … … … (i)

(ii) Equation of

… … … … (ii)

Solving (i) and (ii) we get and .

Therefore is

Question 17: From the given figure, find:

(i) The co-ordinates of .

(ii) The equation of the line through and parallel to . **[2004]**

Answer:

Slope of

The equation of line parallel to and passing through

Question 18: are the vertices of triangle . Write down the equation of the median of the triangle through . **[2005]**

Answer:

Mid point of

Therefore equation passing through and is

Question 19: are vertices of a triangle . If is the mid-point of , use co-ordinate geometry to show that is parallel to . Give a special name to quadrilateral .

Answer:

Coordinates of

Coordinates of

Slope of

Slope of

Therefore .

is a trapezoid.

Question 20: A line meets the at point and at point . The point divides the line segment internally such that . Find:

(i) The co-ordinates of .

(ii) Equation of the line through and perpendicular to .

Answer:

(i) Let and

Therefore

Similarly,

Therefore and

(ii) Slope of

Slope of line perpendicular to

Therefore the equation of line passing through with slope :

Question 21: A line intersects at point and cuts off an intercept of units from the positive side of . Find the equation of the line. **[1992]**

Answer:

Equation of line

Question 22: Find the equation of a line passing through the point and having the of units.** [2002]**

Answer:

Equation of line passing through and

Question 23: The given figure (not drawn to scale) shows two straight lines . If equation of the line and equation of is . Write down the inclination of lines ; also, find the angle between **[1989]**

Answer:

Slope of

Slope of

Therefore

Question 24: Write down the equation of the line whose gradian is and which passes through , where divides the line segment joining in the ratio **[2001]**

Answer:

Given divides the line segment joining in the ratio

Let the coordinates of

Therefore

Equation of a line passing through with slope

Question 25: The ordinate of a point lying on the line joining points . Find the co-ordinates of that point.

Answer:

Let the ordinate of a point lying on the line joining points be

Equation of line passing through and

Therefore if , then

Therefore the point is

Question 26: Point have co-ordinates respectively. Find:

(i) The slope of

(ii) The equation of perpendicular bisector of the line segment

(iii) The value of lies on it **[2008]**

Answer:

(i) Slope of

(ii) Mid point of

Therefore equation of line passing through and slope is

(iii) lies on it

Therefore

Question 27: are two points on the respectively. is the mid-point of . Find the

(i) Co-ordinates of

(ii) Slope of line

(iii) Equation of line **[2010]**

Answer:

Let and

is the mid point

(i) Therefore

Hence and

(ii) Slope of

(iii) Equation of

Question 28: The equation of a line is . Find:

(i) Slope of the line.

(ii) The equation of a line perpendicular to the given line and passing through the intersection of the lines and **[2010]**

Answer:

(i) Slope

(ii) Slope of perpendicular

For point of intersection solve and

and

Therefore intersection

Therefore equation of line

Question 29: is a parallelogram where and . Find: (i) Co-ordinates of (ii) Equation of diagonal **[2011]**

Answer:

(i) Mid point of

Therefore we have and

is the mid point of as well (diagonals of a parallelogram bisect each other)

Hence

and

Hence

(ii) Equation of

Question 30: Given equation of line _{ }is .

(i) Write the slope of line _{ }is is the bisector of angle

(ii) Write the co-ordinates of point .

(iii) Find the equation of

Answer:

(i)

Therefore slope

(ii) Therefore

(iii) Equation of line