Question

Find the coordinates of point P. if P divides the line segment AB joining the points A(2,1) and B(-3,6) in the ratio 2:3.​

Answer 1

The point P lies inside line AB.

So the formula to be used here is Section Formula-Internal Division.

• $\sf{(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n} )}$ where the ratio is $\sf{m:n}$

Given two points are A, B.

Now the formula gives us the point P.

$\sf{P(\dfrac{-3\times2+2\times3}{5}, \dfrac{6\times2+1\times3}{5} )}$

Therefore the point will be $\sf{P(0, 3)}$.

How is the formula made?

On the number line,

let us consider $\sf{A(x_1)}$ and $\sf{B(x_2)}$. ($\sf{x_1<x<x_2}$) ...(1)

In the ratio $\sf{m:n}$, the point $\sf{P(x)}$ will divide line AB. ...(2)

The lengths are:

• $\sf{AP=x-x_1>0}$ ...Because of (1)
• $\sf{BP=x_2-x>0}$ ...Because of (1)

Now $\sf{AP:BP=m:n}$ ...Because of (2)

$\sf{(x-x_1):(x_2-x)=m:n}$

∴$\sf{m(x_2-x)=n(x-x_1)}$

∴The solution to $\sf{P(x)}$ is $\sf{P(\dfrac{mx_2+nx_1}{m+n} )}$.

If we approach with geometry,

we might also know that x and y axes are perpendicular.

If we think outside the box,

we can approach the vertex as a combination of two number lines.

Now the formula becomes $\sf{P(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n} )}$

because vertex has two components.

$\sf{\therefore{P(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n} )}}$

Same for external, the only difference is

• $\sf{x_1<x<x_2}$ [Internal]
• $\sf{x<x_1<x_2}$ or $\sf{x_1<x_2<x}$ [External]

As x should not be inside the interval.

$\sf{\therefore{P(\dfrac{mx_2-nx_1}{m-n}, \dfrac{my_2-ny_1}{m-n} )}}$

The midpoint formula can be found by applying $\sf{1:1}$,

on the internal section formula.

$\sf{\therefore{P(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} )}}$

Answer 2

Answer:

The coordinates or point P are ( 0, 3 ).

Step-by-step-explanation:

We have given the coordinates of the points of a line segment and the ratio in which a point divides the line segment.

We have to find the coordinates of that dividing point.

A ≡ ( 2, 1 ) ≡ ( x₁ , y₁ )

B ≡ ( - 3, 6 ) ≡ ( x₂, y₂ )

P ≡ ( x, y )

m : n = 2 : 3

Now, by Section formula,

x = ( mx₂ + nx₁ ) / ( m + n )

⇒ x = [ 2 ( - 3 ) + 3 ( 2 ) ] / ( 2 + 3 )

⇒ x = ( - 6 + 6 ) / 5

⇒ x = 0 / 5

⇒ x = 0

Now,

y = ( my₂ + ny₁ ) / ( m + n )

⇒ y = [ 2 ( 6 ) + 3 ( 1 ) ] / ( 2 + 3 )

⇒ y = ( 12 + 3 ) / 5

⇒ y = 15 / 5

⇒ y = 3

∴ P ≡ ( x, y ) ≡ ( 0, 3 )

∴ The coordinates or point P are ( 0, 3 ).

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Additional Information:

1. Distance Formula:

The formula which is used to find the distance between two points using their coordinates is called as distance formula.

• d ( A, B ) = √[ ( x₁ - x₂ )² + ( y₁ - y₂ )² ]

2. Section Formula:

The formula which is used to find the coordinates of a point which divides a line segment in a particular ratio is called as section formula.

• x = ( mx₂ + nx₁ ) / ( m + n )

• y = ( my₂ + ny₁ ) / ( m + n )

3. Midpoint Formula:

The formula which is used to find the coordinates of the midpoint of a line segment is called the midpoint formula.

• x = ( x₁ + x₂ ) / 2

• y = ( y₁ + y₂ ) / 2