Question 1: Point divides the line segment joining the points
in the
Find the co-ordinates of point
Also, find the equation of the line through
and parallel to
Answer:
divides
in the
Therefore
Therefore the required equation is
Question 2: The line segment joining the points is divided in the
at point
in it. Find the co-ordinates of
Also, find the equation of the line through
and perpendicular to the line
Answer:
in the
Therefore the required equation is
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
Question 4: Find the value of for which the lines
are perpendicular to each other. [2003]
Answer:
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 the line
The co-ordinates of
The co-ordinates of [2003]
Answer:
The equation of the line:
Question 6: are the co-ordinates of vertices
respectively of rhombus
Find the equations of the diagonals
Answer:
Equation of :
:
Question 7: Show that can be vertices of a square. Find the coordinates of its fourth vertex
, if
is a square. Without using the coordinates of vertex
, find the equation of side
of the square and the equation of diagonal
Answer:
In a square, the diagonals bisect each other. Therefore
:
:
Question 8: A line through origin meets the line at right angles at point
find the coordinates of point
Answer:
.. … … … (i)
The equation of a line passing through and having slope
is
.. … … … (i)
Solving equations (i) and (ii)
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
is the
Therefore:
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
in (i) we get
Therefore the intercept is
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:
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:
Therefore the equation of median of through
is
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:
Therefore line passing through is perpendicular to
in
we get that it satisfies the equation.
bisects the line joining
Question 14: Given a straight line Determine the equation of the other line which is parallel to its and passes through
Answer:
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:
(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 coordinates of the point , where the perpendicular through
, as obtained in (i.), meets
Answer:
Therefore equation of line passing through with slope
is:
.. … … … (i)
(ii) Equation of
.. … … … (ii)
Solving (i) and (ii) we get
Question 17: From the given figure, find:
(i) The co-ordinates of
(ii) The equation of the line through and parallel to
[2004]
Answer:
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:
Therefore equation passing through 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:
is a trapezoid.
Question 20: A line meets the
at point
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:
Similarly,
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
Question 23: The given figure (not drawn to scale) shows two straight lines If equation of the line
and equation of
Write down the inclination of lines
; also, find the angle between
[1989]
Answer:
Question 24: Write down the equation of the line whose gradian is and which passes through
, where
divides the line segment joining
in the
[2001]
Answer:
divides the line segment joining
in the
Let the coordinates of
Therefore
Question 25: The ordinate of a point lying on the line joining points Find the co-ordinates of that point.
Answer:
Equation of line passing through
Therefore if , then
Therefore the point is
Question 26: Point have co-ordinates
respectively. Find:
(i) The
(ii) The equation of perpendicular bisector of the line segment
(iii) The value of lies on it [2008]
Answer:
Therefore equation of line passing through and slope
is
Question 27: are two points on the
respectively.
is the mid-point of
Find the
(i) Co-ordinates of
(iii) Equation of line [2010]
Answer:
is the mid point
(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 [2010]
Answer:
For point of intersection solve
Therefore equation of line
Question 29: is a parallelogram where
Find: (i) Co-ordinates of
(ii) Equation of diagonal
[2011]
Answer:
is the
as well (diagonals of a parallelogram bisect each other)
(ii) Equation of
Question 30: Given equation of line
(i) Write the is the bisector of angle
(ii) Write the co-ordinates of point
(iii) Find the equation of
Answer:
Therefore slope
(iii) Equation of line