*Note: Equation of a like is also defined as *

Question 1: Find the equation of the straight through the following pair of points:

i) and ii) and

iii) and iv) and

v) and vi) and

Answer:

i) Given points and

Slope of

Substituting in we get the equation of as:

ii) Given points and

Slope of

Substituting in we get the equation of as:

iii) Given points and

Slope of

Substituting in we get the equation of as:

iv) Given points and

Slope of

Substituting in we get the equation of as:

v) Given points and

Slope of

Substituting in we get the equation of as:

vi) Given points and

Slope of

Substituting in we get the equation of as:

Question 2: Find the equation to the sides of the triangles the coordinates of whose angular points are respectively: (i) and ii) and

Answer:

i) Given points and

Slope of

Therefore equation of :

Slope of

Therefore equation of :

Slope of

Therefore equation of :

ii) Given points and

Slope of

Therefore equation of :

Slope of

Therefore equation of :

Slope of

Therefore equation of :

Question 3: Find the equations of the medians of a triangle, the the coordinates of the vertices are and .

Answer:

Given points and . Please refer to the figure shown.

Let be the mid point of and respectively. Therefore,

Slope of

Therefore equation of :

Slope of

Therefore equation of :

Slope of

Therefore equation of :

Question 4: Find the equations to the diagonals of the rectangle the equations of whose sides are and .

Answer:

Please refer to the figure shown. Therefore the four points are

Slope of

Therefore equation of :

Slope of

Therefore equation of :

Question 5: Find the equation of the side of the whose vertices are and respectively. Also, find the equation of the median through .

Answer:

Given points and

Slope of

Therefore equation of :

Mid point of

Slope of

Therefore equation of :

Question 6: Using the concept of the equation of a line, prove that the three points and are collinear.

Answer:

Given points and

Slope of

Therefore equation of :

Now check if satisfy the equation

Therefore are collinear.

Question 7: Prove that the line , divides the join of points and in the ratio .

Answer:

Let , divides the join of points and at a point P in the ratio of

Since lies on , it should satisfy the equation.

Hence the line , divides the join of points and in the ratio .

Question 8: Find the equation to the straight line which bisects the distance between the points and also bisects the distance between the points and .

Answer:

Given points and and

Mid point of

Mid point of

Slope of

Therefore equation of :

Question 9: In what ratio is the line joining the points and divided by the line passing through the points and .

Answer:

Given points: and

Slope of

Therefore equation of :

Let divides the line joining and in the ratio of

point of intersection

This point is on line

Hence the required ratio is externally.

Question 10: The vertices of a quadrilateral are and . Find the equations of its diagonals.

Answer:

Given points:

Slope of

Therefore equation of :

Slope of

Therefore equation of :

Question 11: The length (in centimeters) of a copper rod is a linear function of its Celsius temperature . In an experiment, if when , and when , express in terms of .

Answer:

Given two points and

Slope of line

Therefore the equation of line:

Question 12: The owner of a milk store finds that he can sell liters milk each week at ( per liter and liters of milk each week at Rs. per liter. Assuming a linear relationship between selling price and demand, how many liters can he sell weekly at Rs. per liter.

Answer:

Given two points and

Slope of line

Therefore the equation of line:

When

Hence the owner of the milk store would be able to sell liters of milk at Rs. / liter.

Question 13: Find the equation of the bisector of of the triangle whose vertices are and .

Answer:

Let be the bisector of .

We know,

Now

Therefore coordinates of

Slope of

Therefore equation of :

Question 14: Find the equations to the straight lines which go through the origin and trisect the portion of the straight line which is intercepted between the axes of coordinates

Answer:

Given line

x-intercept

y-intercept

divides in the ratio of

Coordinates of

divides in the ratio of

Coordinates of

Slope of

Therefore equation of :

Slope of

Similarly, equation of :

Question 15: Find the equations of the diagonals of the square formed by the lines and .

Answer:

Slope of

Therefore equation of :

Slope of

Similarly, equation of :