7. Coordinate Geometry
Coordinate Geometry
7. COORDINATES GEOMETRY
SECTION = A
Q1. The point 
is equidistant from the points 
and 
,then : [ SEBA 2020]
(a) 
(b) 
(c) 
(d) 
Solution: (c) .
[ let the distance of 
from 
and 
are equal .
A/Q, 
]
Q2. The distance between the points 
and 
is : [SEBA 2019]
(a) 10 units
(b) 8 units
(c) 6 units
(d) 2 units
Solution: (b) 10
[ The distance between the points 
units ]
Q3. If 
is mid-point of the line segment joining the points 
and 
,then 
is :
(a) – 1 (b) 1 (c) 2 (d) – 2
Solution: (b) 1
[ We have , 
]
Q4. The distance between the points 
and 
is : [SEBA 2015 , 18]
(a) 2 (b) 
(c) 1 (d) 0
Solution: (b)
[ The distance between the points 
unit ]
Q5. The line segment joining the points 
and 
in the ratio 2 : 3 , then the coordinate of the point is :
(a) 
(b) 
(c) 
(d) 
Solution: (d) .
[ Here, 
, 
, 
, 
, 
and 
and 
]
Q6. The distance between the point 
and 
is :
(a) 
(b) 
(c) 
(d) 
Solution: (c)
[ The distance 
unit ]
Q7. The distance of the point 
from X-axis is : [SEBA 2017]
(a) 2 (b) 5 (c) 1 (d) 3
Solution: (d) 3 units
Q8. The mid-point of the line segment joining the points 
and 
is 
, then 
is : [SEBA 2016]
(a) – 4 (b) – 12 (c) 12 (d) – 6
Solution: (b) – 12
[ We know that, the coordinates of the mid-point 
of the
join of the points 
and 
is 
.
A/Q, 
Q9. The distance between the point 
and 
is :
(a) 
(b) 
(c) 
(d) 
Solution: (c) 
.
[ Using the distance formula , we have
unit ]
Q10. The distance between the points P
and Q
is : [ CBSE 2020 Basic]
(a) 
units (b) 
units (c) 
units (d) 40 units
Solution: (c) units
[ Using the distance formula , we have
units ]
Q12. The point on the x-axis which is equidistant from 
and 
is :[ CBSE 2020 Standard ]
(a) 
(b) 
(c) 
(d) 
Solution: (d) .
[ Since the point 
is equidistant from and
So, 
Therefore, the point is 
. ]
Q13. The centre of a circle whose end points of a diameter are 
and 
is : [ CBSE 2020 Standard ]
(a) 
(b) 
(c) 
(d) 
Solution: (c) .
[ let the point 
is the midpoint of and .
A/Q , 
]
Q14. If the points 
and 
lie on the y-axis , then the distance of CD is :
(a) 
units (b) 8 units (c) 3 units (d) 5 units
Solution: (d) 5 units .
[ The distance 
units ]
Filled in the blanks
Q1. The coordinates of the point which divides the join of 
and 
in the ratio 
is 
.
Solution: 
.
[ Here, 
, 
,
, 
, 
and 
and 
]
Q2. If the coordinates of one end of a diameter of a circle are 
and the coordinates of its centre are 
, then the coordinate of the other end of the diameter is 
. [CBSE 2012]
Solution: 
.
[ Since the point is the midpoint of and .
A/Q , 
and 
]
Q3. The point which lies on the perpendicular bisector of the line segment joining the points 
and 
is 
.
Solution: 
.
[ let the point is the midpoint of 
and 
.
A/Q , 
Q4. If P and Q be the points of trisection of the line segment joining the points 
and 
such that P is nearer to A , then the ratio of 
.
Solution: 2 : 1
[ The ratio of the points and is 2 : 1 . ]
Q5. If the distance between the points 
and 
is 5,then 
is .
Solution: 0
[ Since the distance between the points 
and 
is 5 .
A/Q , 
]
Q6. The line segment joining 
and 
is divided by the -axis , then the ratio is .
Solution: 1 : 1
[ Given, -axis , i.e., 
.
A/Q, 
]
Q7. If the points 
are collinear , then the value of is 
. [ CBSE 2014]
Solution: – 63
[ Here , 
, 
, 
, 
, 
, 
We know that , 
]
Answers of the following Questions
Q1. Find the mid-point of the line segment joining the points 
and 
.
Solution: We know that , 
Therefore, the mid-point of the line segment is 
.
Q2. If 
is the mid-point of the line segment joining 
and 
,then find 
.
Solution: Here , 
, 
, 
, 
, 
, 
We know that , 
Q3. Find the distance of a point 
from the origin .
Solution: Given , the distance of the point and 
is
Q4. Find the distance between the points and 
.
Solution: Given , the distance of the point 
and 
is
units
Q5. If the distance between the points 
and is 5 , then find the value of 
. [CBSE 2017 ]
Solution : Given, the distance between the points 
and 
is 5 .
A/Q , 
SECTION = B
Q1. Find the perimeter of a triangle with vertices 
, and . [CBSE 2014 F]
Solution: Let 
,
and 
are the vertices of the triangle.
The perimeter of 
units
Q3. Find the point on the -axis which is equidistant from 
and 
. [SEBA19]
Solution: Let 
is equidistant from 
and .
According to question, 
Therefore , the point is 
.
Q4. The -coordinate of a point P is twice its 
-coordinate . If P is equidistant from and 
, find the coordinates of P . [2016D]
Solution : Let the coordinate of the point P is .
Given, the -coordinate of a point P is twice its -coordinate , i.e., 
.
A/Q, 
.
So, the coordinate of the point P is 
.
Q5. Prove that the points 
, 
and are the vertices of a right angled isosceles triangle . [CBSE 2016]
Solution: Let the points , 
and 
are the vertices of the triangle.
units
units
units
So, ABC is an isosceles triangle .
Therefore, 
units and 
units
So, ABC is a right angled isosceles triangle . Proved.
Q6. If 
, 
, 
and 
are the vertices of a parallelogram taken in order, find and .
Solution: We know that diagonals of a parallelogram bisect each other .
So, the coordinates of the mid-point of AC = the coordinates of the mid-point of BD .
and 
and 
Q7. Find the value of 
if the points 
, 
and 
are collinear .
Solution: Here , 
, 
, 
, 
, 
, 
We know that ,
Q8. If the distance of from 
and 
are equal , then prove that 
.
Solution: Given , the distance of from and are equal .
According to question, 
Proved.
Q9. If the point is equidistant from the points 
and 
,prove that 
. [CBSE 2017C]
Solution: Since, 
is equidistant from the points A and B 
.
A/Q , 
Proved.
Q10. Find a relation between and such that the point is equidistant from the point 
and 
.
Solution: Given ,the point 
is equidistant from the point 
and 
.
A/Q , 
[ Squaring both side]
SECTION = C
Q1. If A
, 
, C 
and D
are the vertices of a quadrilateral ,find the area of the quadrilateral ABCD .
Solution: Since A , 
, 
and 
are the vertices of quadrilateral ABCD .
Join BD and we find ABD and BCD are two triangle .
sq. unit
and 
sq. unit
Therefore, area of 
Sq.units 
sq. unit
Q2. If 
, 
and 
are the vertices of a right angled triangle with 
, then find the value of 
. [Delhi 2015]
Solution:
Since, A , and are the vertices of a right angled triangle .
Q3. Find the ratio in which the point 
divides the line segment joining the points 
and 
. Also , find the value of . [CBSE 2016]
Solution : let the ratio is 
.
Here, 
, 
, 
, 
, 
and 
Using section formula , we have
and 
Therefore, the ratio is 2 : 1 and the value of is 
.
Q4. Find the ratio in which the 
-axis divides the line segment joining the points 
and 
. Also, find the point of intersection .
Solution: Solution: let the ratio is and the coordinates of the point is 
.
Here , 
, 
, 
, 
Using section formula , we have
and
Therefore, the ratio is 5 : 1 and the coordinates of the point is 
.
Q5. If A 
, B 
, C
and D are the vertices of a parallelogram ABCD, find the values of 
and . Hence find the lengths of its sides . [CBSE 2018]
Solution: We know that diagonals of a parallelogram bisect each other .
So, the coordinates of the mid-point of AC = the coordinates of the mid-point of BD .
and 
Thus, A , B , C
and D are the vertices of a parallelogram ABCD .
units
units
units
units
Q6. Find the ratio in which the point of intersection of the -axis and the line segment which joins the points and 
internally divides the line segment . Also, find the coordinates of the point . [SEBA 2014]
Solution: let the ratio is and the coordinates of the point is .
Here , 
,
, 
, 
Using section formula , we have
and
Therefore, the ratio is 5 : 7 and the coordinates of the point is 
.
Q7. If A and B are 
and 
respectively , find the coordinates of P such that 
and P lies on the line segment AB .
Solution: let, the coordinate of P is 
Here, , 
, 
, 
, 
and
Using section formula, We have
and
Therefore, the coordinate of P is 
.
Q8. Determine the ratio in which the line 
divides the line segment joining the points A
and B 
. Also, find the coordinate of the points .
Solution: let the ratio is and given the coordinate is .
Here, , , 
and 
Using section formula, We have
and 
A/Q , 
and 
Therefore, the ratio is 2 : 9 and the coordinate of the point is 
.
Q9. If A , B
, C
and D
are the vertices of a quadrilateral ABCD of area 80 Square units , then find positive value of .
Solution: Given, ABCD be a quadrilateral and BD join .
ar(
) 
Sq. units (positive value)
and ar(
) 
sq. units (positive value)
A/Q , 
Therefore, the positive value of is 8 .
Q10. Show that the points 
, 
, 
and 
are the vertices of a square .
Solution: let, 
and 
are the given points.
So, 
and 
.
Therefore,
is a square .
SECTION = D
Q1. In figure 6, ABC is a triangle coordinates of whose vertex A is 
. D and E respectively are the mid-points of the sides AB and AC and their coordinates are and 
respectively . If F is the mid-point of BC , find the areas of 
and 
.
Solution: let 
and 
are the coordinates of the triangle ABC .
Since D be a mid-point of AB . So , 
and 
Since E be a mid-point of AC . So , 
and 
Therefore, the coordinates of triangle B and C are 
and 
.
The vertices of the triangle ABC are 
, 
and 
.
sq. unit
Again , F is a mid-point of BC , then the coordinate of F is 
,i.e. (1 , 2) .
The vertices of the triangle DEF are 
,
and 
.
sq. unit ( Area always positive)
Posted 5 years ago