If A and B are (− 2, − 2) and (2, − 4), respectively, find the coordinates of P such that AP=37AB and P lies on the line segment AB. - Mathematics

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Sum

If A and B are (− 2, − 2) and (2, − 4), respectively, find the coordinates of P such that `AP = 3/7 AB` and P lies on the line segment AB.

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Solution 1

The coordinates of point A and B are (−2, −2) and (2, −4) respectively.

Since AP = `3/7 AB`

Therefore, AP: PB = 3:4

Point P divides the line segment AB in the ratio 3:4.

Coordinates of P = `((3xx2+4xx(-2))/(3+4), (3xx(-4)+4xx(-2))/(3+4))`

`= ((6-8)/7, (-12-8)/7)`

`=(-2/7, -20/7)`

Solution 2

We have two points A (-2,-2) and B (2,-4). Let P be any point which divides AB as

`AP = 3/7 AB`

Since,

AB = (AP + BP)

So,

7AP = 3AB

7AP = 3(AP + BP)

4AP = 3BP

`(AP)/(BP) = 3/4`

Now according to the section formula if any point P divides a line segment joining `A(x_1, y_1)` and `B(x_2, y_2)` in the ratio m: n internally than,

`P(x,y) = ((nx_1 + mx_2)/(m + n)"," (ny_1 + my_2)/(m + n))`

Therefore P divides AB in the ratio 3: 4. So,

`P(x,y) = ((3(2) + 4(-2))/(3 + 4)"," (3(-4) + 4(-2))/(3 + 4))`

`= (-2/7,-20/7)`

Concept: Section Formula
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Chapter 7: Coordinate Geometry - Exercise 7.2 [Page 167]

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NCERT Mathematics Class 10
Chapter 7 Coordinate Geometry
Exercise 7.2 | Q 8 | Page 167
RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.3 | Q 39 | Page 30
RS Aggarwal Secondary School Class 10 Maths
Chapter 16 Coordinate Geomentry
Exercises 2 | Q 3
RD Sharma Class 10 Maths
Chapter 9 Constructions
Exercise 9.3 | Q 39 | Page 30

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