Note: We are going to use the following formula extensively in solving the following problems. The distance between any two points is
Notes: If a point is on , its ordinate is
, therefore the point on
is taken as
. Similarly, if the point is on
, its abscissa is
, therefore the point on
is taken as
. For details refer to the following lecture notes.
Question 1: Find the distance between the following pairs of points:
Answer:
and (
Question 2: Find the distance between the origin and the points:
Answer:
Question 3: The distance between point is
. Find
Answer:
Question 4: Find the coordinate of the point on which are at a distance of
units from the point
Answer:
Therefore the points are
Question 5: Find the coordinate of the point on which are at a distance of
units from the point
Answer:
Hence the points could be
Question 6: A point is at a distance of
units from the point
. Find the coordinates of the point
if its ordinate is twice its abscissa.
Answer:
Question 7: A point is equidistant from the point
. Find
Answer:
is equidistant from
Therefore
Question 8: What point on is equidistant from the point
Answer:
Let the point be . Therefore
Therefore the point is
Question 9: What point on is equidistant from the point
Answer:
Let the point be . Therefore
Hence the point is
Question 10: A point lies on
and another point
lies on
. Write the ordinate of point
, abscissa of point
. If the abscissa of point
is
and ordinate of point
is
. Calculate the length of the line segment
Answer:
Let . Given
Question 11: Show that the points are the vertices of an isosceles triangle.
Answer:
Therefore two sides are equal which makes it an isosceles triangle.
Question 12: Prove that the points are the vertices of the rectangle
Answer:
Therefore .
Hence it is a rectangle.
Question 13: Prove that the points are the vertices of an isosceles triangle. Find the area of the triangle.
Answer:
For this to be a right angled triangle we should have
. Hence proved that it is a right angled triangle.
Question 14: Show that the points are the vertices of the square
Answer:
Therefore .
Hence it is a square.
Question 15: Show that are the vertices of a rhombus.
Answer:
Therefore .
Two sides are equal and the other two sides are of different length. Hence it is a rhombus.
Question 16: Points are the vertices of a quadrilateral
. Find a if a is negative and
Answer:
Therefore
. Hence
as it is negative.
Question 17: The vertices of a triangle are . Find the coordinates of the circumcenter of the triangle.
Answer:
are the points
Let the coordinates of the circumcenter
Therefore
Hence
Therefore equation 1:
Also equation 2:
Hence the coordinates of the circumcenter is
Question 18: Given . Find
if
Answer:
Therefore
Question 19: Given . Find
if
Answer:
Question 20: The center of the circle is . Find
if the circle passes through
and the length of the diameter is
units.
Answer:
Diameter units i.e.Radius
units
Question 21: The length of the line is
units and the coordinates of
are
, calculate the coordinates of point
, if its abscissca is
Answer:
and Let
Hence the points could be
Question 22: Point is the center of the circle with radius
units,
is perpendicular to chord
. Calculate the length of
Answer:
; Radius
units;
Therefore
Question 23: Calculate the distance between the two points , correct to three significant figures. [1990]
Answer:
Question 24: Calculate the distance between on the
whose abscissa is
[1997]
Answer:
Question 25: Calculate the distance between on the
whose ordinate is
Answer:
Question 26: Find the point on whose distance from the point
are in the ratio of
Answer:
Let point be
Hence points could be
Question 27: The distance of point from the points
are in the ratio
. Show that:
Answer:
. Hence proved.
Question 28: The point are vertices of triangle
right angles at vertex
. Find the value of
Answer: