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: