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 :