7. Coordinates Geometry

SEBA Class 10 Maths Chapter 7. Coordinates Geometry

Chapter 7. Coordinates Geometry

Class 10 Maths Chapter 7 Coordinate Geometry Multiple Questions and Solutions :

Question : The mid-point of the line segment joining the points (5,1) and (m , 3) is (2,2) , then the value of m is : [SEBA2013]

(a) – 2 (b) – 1 (c) 2 (d) 1

Solution : (b) – 1

[ Using section formula , we have

and

Now,

]

Question : The distance of the point (– 3 ,4) from the origin is : [SEBA 2014]

(a) 1 (b) 7 (c) 12 (d) 5

Solution : (d) 5

[ let the distance of the point P(– 3 ,4) from O(0,0)

Using distance formula, we have

]

Question : The ratio in which y-axis divides the line segment joining the points (– 2,0) and (4 , 0) is : [SEBA 2023]

(a) 2:3 (b) 1:2 (c) 1:4 (d) 2:1

Solution : (b) 1:2

[ let, the ratio be

Given, the point on the y-axis , i.e.,

Here, , , ,

Using section formula ,

]

Question : 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 ,

]

Question : The distance between the points and is : [SEBA 2019]

(a) 10 units (b) 8 units (c) 6 units (d) 2 units

Solution: (b) 10 units

[ The distance between the points

units ]

Qusetion : 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,

]

Question : The distance between the points and is : [SEBA 2015 ,2018]

(a) 2 (b) (c) 1 (d) 0

Solution: (b)

[ The distance between the points

unit ]

Question : 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 ]

Question : The distance between the point and is :

(a) (b) (c) (d)

Solution: (c)

[ The distance

unit ]

Question : The distance of the point from X-axis is : [SEBA 2017]

(a) 2 (b) 5 (c) 1 (d) 3

Solution: (d) 3 units

Question : The ratio in which the -axis divides the line segment joining the points and ( is :

(a) 1 : 2 (b) 3 : 2 (c) 2 : 1 (d) 2 : 3

Solution: (a) 1 : 2 .

[ Let the ratio is .

Here , , , and .

Given, -axis , i.e., .

A/Q,

]

Question : 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,

]

Question : The distance between the point and is : [SEBA 2021]

(a)

(b)

(c)

(d)

Solution: (c) .

[ Using the distance formula , we have

unit ]

Question: The point which divides the line segment joining the points and in ratio internally lies in the –

(a) quadrant (b) quadrant (c) quadrant (d) quadrant

Solution: (c) quadrant [ Here, , , , , ,

Using section,

and

The point is . ]

Question : 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 ]

Question : The mid-point of the line segment joining the points and is :

(a) (b) (c) (d)

Solution: (d)

[ We know that ,

Therefore, the mid-point of the line segment is .

Question : If is the mid-point of the line segment joining and ,then is :

(a) (b) 1 (c) 2 (d)

Solution: (a)

[ Here , , , ,, ,

We know that ,

]

Question : The distance of a point from the origin is (in units) :

(a) (b) (c) (d)

Solution: (b)

[ Given , the distance of the point and is

units ]

Question: The distance between the points and is (in units) :

(a) 36 (b) 40 (c) 64 (d) 39

Solution: (d) 39

[ Given , the distance of the point and is

units ]

Question : The line segment joining the points and is divided by P such that , then the ratio of is :

(a) 4 : 3 (b) 3 : 2 (c) 2 : 3 (d) 3 : 4

Solution: (d) 3 : 4

[ We have,

]

Question : 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 . ]

Question: 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,

]

Question : If the distance between the points and is 5 , then the value of is : [2017D]

(a) 4 (b) (c) (d) 5

Solution : (c)

[ Given, the distance between the points and is 5 .

A/Q,

]

Question : The perimeter of a triangle with vertices , and is : [2014 F]

(a) 6 units (b) 10 units (c) 12 units (d) 14 units

Solution: (c) 12 units

[ Let , and are the vertices of the triangle.

The perimeter of

units ]

Question : 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 ]

Question : The middle point of the line segment joining the points and is , then the value of is :

(a) – 8 (b) 1 (c) 7 (d) 8

Solution: (d) 8

[We know that ,

]

Question: If the distance between the points and is 5 , then is .

(a) 5 (b) 0 (c) (d) 25

Solution: (b) 0

[ Since the distance between the points and is 5

A/Q ,

]

Class 10 Maths Chapter 7. Coordinate Geometry Filled in the blanks

Question : The coordinates of the point which divides the join of and in the ratio is .

Solution: .

[ Here, , , , , and

and ]

Question : 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

]

Question : 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 ,

Question : 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 . ]

Question : If the distance between the points and is 5,then is .

Solution: 0

[ Since the distance between the points and is 5 .

A/Q ,

]

Question : The line segment joining and is divided by the -axis , then the ratio is .

Solution: 1 : 1

[ Given, -axis , i.e., .

A/Q,

]

Question : If the points are collinear , then the value of is . [ CBSE 2014]

Solution: – 63

[ Here , , , , , ,

We know that ,

]

Class 10 Coordinate Geometry 2 Marks Questions and Answers SECTION = B

Question : Find the values of for which the distance between the points and is 10 units .

Solution : We have ,

So, the value of are – 9 and 3 .

Question : 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 .

Question : In what ratio does the point (– 4, 6) divide the line segment joining the points A(– 6, 10) and B(3, – 8)?

Solution: Let the ratio is .

Here, , ,, ,,

Using section formula,

and

Now,

Therefore, the ratio is .

Class 10 Coordinate Geometry 3 Marks Questions and Answers SECTION = C

Question : Show that the points (1, 7), (4, 2), (–1, –1) and (– 4, 4) are the vertices of a square.

Solution : Let A(1, 7), B(4, 2), C(–1, –1) and D(– 4, 4) be the given points.

Using distance formula , we have

So, and

Thus , ABCD is a square .

Question : Find a point on the y-axis which is equidistant from the points A(6, 5) and B(– 4, 3).

Solution : Given, the point on the x-axis .

Let the point be P( ) .

A/Q ,

Therefore, the point is (0 , 9) .

Question : Check whether and are the vertices of an isosceles triangle .

Solution : Let and are the vertices of any triangle respectively .

Using distance formula , we have

units

So,

Therefore ,the points and are the vertices of an isosceles triangle .

Question : 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]

Question : Find the value of k if the points and are collinear.

Solution: Here, ,

We know that ,

Therefore, the value of k is 4 .

Question : Find the ratio in which the line segment joining the points and is divided by .

Solution: let , the ratio be .

Here, , ,

Using section formula , we have

and

Therefore, the ratio is 2 : 7 .

Question : Find the ratio in which the line segment joining and is divided by the -axis . Also , find the coordinates of the point of division .

Solution: let , the ratio be and the coordinate is .

Here, ,

Using section formula , we have

and

Now,

Again,

Therefore, the ratio is 1 : 1 and the coordinate is .

Question : If is equidistant from and , find the values of . Also find the distance QR and PR .

Solution : Since is equidistant from and .

A/Q,

The distance of and is

units

The distance of and is

units

The distance of and is

units

The distance of and is

units

Question : 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 .

Question : 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

So, ABC is an isosceles triangle .

Therefore, units and units

So, ABC is a right angled isosceles triangle . Proved.

Question : 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

Question : Find the value of if the points , and are collinear .

Solution: Here , , , , , ,

We know that ,

Question : If the distance of from and are equal , then prove that .

Solution: Given , the distance of from and are equal .

A/Q ,

Proved.

Question : 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]

Question : 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

Question : 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 .

Question : Find the ratio in which the -axis divides the line segment joining the points and . Also, find the point of intersection .

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 .

Question : 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 .

Question : Find the coordinates of a point A , where AB is the diameter of a circle whose centre is and is .

Solution: Here, AP = BP = Radius . So,

Let the coordinates of a point .

Using the section, we have

and

Therefore, the coordinate of the point .

Question: 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 coordinates of the point P is .

And

Here, , ,

Using section formula , we have

and

Therefore, the coordinate of the point P is .

Question : 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 ,

Here, m = 2 , n = 9

and

Therefore, the ratio is 2 : 9 and the coordinate of the point is .

Question : 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 .

Class 10 Coordinate Geometry 4 Marks Questions and Solutions

SECTION = D

Question : 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)