Question

let a(2,2),b(8-4) be to given points in a plane if a point p lies on the X-axis (in toe) and divides AB in the ratio 1:2 then find the coordinates of P​

Answer 1

Answer:

P(4, 0) are it's coordinates.

And PB : AP = 2 : 1

Step-by-step explanation:

A(2, 2), B(8, -4)

Equation representing line AB:

$\frac{y - 2}{2 - (-4)} = \frac{x - 2}{2 - 8}$

$\frac{y - 2}{6} = \frac{x - 2}{-6}$

$=> y - 2 = -x + 2$

$=> x + y - 4 = 0$  . . . . . . . . equation (i)

Point P lies on the X-axis and divides AB in the ratio 1:2

On the X-axis all points have y coordinate as 0.

Putting this value of y into equation (i):

x + 0 - 4 = 0

∴ x = 4

∴ P(4, 0) are it's coordinates.

Let's verify how P(4, 0) divides line AB:

$AP^2 = (2 - 4)^2 + (2 - 0)^2 = 4 + 4 = 8$

$PB^2 = (4 - 8)^2 + [0 - (-4)]^2 = 16 + 16 = 32$

∴ $\frac{AP^2}{PB^2} = \frac{8}{32} = \frac{1}{4}$

∴ $\frac{AP}{PB} = \frac{1}{2}$

∴ PB : AP = 2 : 1

Answer 2

Concept:

The coordinate of a point that splits a line segment connecting two points into two portions with a length ratio of m:n is found using the section formula.

Assuming that P and Q are the two points (x1,y1) and (x2,y2) that are provided, respectively, and that M is the point that internally divides the line-segment PQ in the ratio m:n, the sectional formula for calculating the coordinate of a point M is as follows:

x= (mx₂+nx₁)/(m+n)

y= (my₂+ny₁)/(m+n)

Given:

Let a(2,2),b(8-4) be to given points in a plane if a point p lies on the X-axis (in toe) and divides AB in the ratio 1:2

Find:

Find the coordinates of P​

Solution:

Using section formula,

x= (mx₂+nx₁)/(m+n)

y= (my₂+ny₁)/(m+n)

m:n=1:2

Let the point P be (x,y)

x=(1 x 8 + 2x2)/3

  =4

y=(1x-4+2x2)/3

 =0

So, the point dividing AB in the ratio of 1:2 is (4,0)

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