Question

find the coordinates of the point which Divides the Join of points.(3,4)and(5,6)and the ratio 1:2​

Answer 1

Given :

Line segment, coordinates of its ends = (3,4) and (5,6)

To Find :

Coordinates of point which divides the line segment in ratio 1:2

Solution :

Let ends of line segment be P(3,4) and Q(5,6)

And point dividing it in ratio 1:2 be R(x,y)

We know that according to section formula,

$\large{\underline{\boxed{\bf{(x,y) = \Bigg(\dfrac{m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m_{2}},\dfrac{m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m_{2}}}}}}$

Finding first coordinate x,

$\\ :\implies\:\sf x = \dfrac{m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m_{2}}$

We have,

x₁ = 3

x₂ = 5

m₁ = 1

m₂ = 2

Putting all values,

$\\ :\implies\:\sf x = \dfrac{1(5) + 2(3)}{1 + 2}$

$\\ :\implies\:\sf x = \dfrac{(1\:\times\:5) + (2\:\times\:3)}{3}$

$\\ :\implies\:\sf x = \dfrac{5 + 6}{3}$

$\\ :\implies\:\sf x = \dfrac{11}{3}$

Finding second coordinate y,

$\\ :\implies\:\sf x = \dfrac{m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m_{2}}$

We have,

y₁ = 4

y₂ = 6

m₁ = 1

m₂ = 2

Putting all values,

$\\ :\implies\:\sf x = \dfrac{1(6) + 2(4)}{1 + 2}$

$\\ :\implies\:\sf x = \dfrac{(1\:\times\:6) + (2\:\times\:4)}{3}$

$\\ :\implies\:\sf x = \dfrac{6 + 8}{3}$

$\\ :\implies\:\sf x = \dfrac{14}{3}$

Hence, (x,y) = (11/3, 14/3)

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Answer 2

Explanation:

Here, O(A)=m and O(B)=n.

Hence O(A×B)=mn

Since every subset of A×B is a relation from A to B, therefore, number of relations from A to B is equal to the number of the subsets of A×B, i.e.,  2mn