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