Question

Find the coordinates of the point which divides the join of (-1,7) and (4,-3) in the ratio 2:3

Answer 1

The coordinates of the point in question is (1, 3).

Explanation:

Point (-1, 7) is above and to the left of the point (4, -3). The point in question is to the right and below the point (-1, 7).

What will be the horizontal distance between the point (-1, 7) and the point in question?

The horizontal distance between the point (-1, 7) and (4, -3) is 5. Let the horizontal distance between the point (-1, 7) and the point in question be
`a`. Let the horizontal distance between the point in question and point (4, -3) be
`b`.

`\displaystyle (a)/(b) = (2)/(3)`.

`\displaystyle a = (2)/(3) \; b`.

`\displaystyle b = (3)/(2)\; a`.

However,

`a + b = 5`.

`\displaystyle a + (3)/(2)\; a = 5`.

`\displaystyle (5)/(2)\; x= 5`.

`a = 2`.

In other words, the point in question is 2 units to the right of the point (-1, 7). The x-coordinate of this point shall be
`-1 + 2 = 1`.

The vertical distance between the point (-1, 7) and the point (4, -3) is 10. Similarly, the point in question is
`(2/5) * 10 = 4` units below the point (-1, 7). The y-coordinate of this point will be
`7 - 4 = 3`.

Thus, the point in question is (1, 3).

Answer 2

To solve our given problem we will use section formula :]

Section Formula states that, when a point divides a line segment internally in the ratio m:n, So the coordinates are :]

`\tiny: \implies (x,y) = \bigg \lgroup x = \frac{m. {x}_(2) +n. {x}_(1) }{m + n} ,y= \frac{m. {y}_(2) +n. {y}_(1) }{m + n} \bigg \rgroup \\ \\ \\`

Let

(-1 , 7) = (x₁ , y₁)

(4 , -3) = (x₂ , y₂)

m = 2

n = 3

• Upon Substituting coordinates of our given points in section Formula we get :]

`\tiny: \implies (x,y) = \bigg \lgroup x = (2 * 4 +3 * - 1 )/(2 + 3) ,y= (2 * - 3 +3 * 7)/(2 + 3) \bigg \rgroup \\ \\ \\`

`\tiny: \implies (x,y) = \bigg \lgroup x = (8 - 3 )/(2 + 3) ,y= ( - 6 +21)/(2 + 3) \bigg \rgroup \\ \\ \\`

`\tiny: \implies (x,y) = \bigg \lgroup x = (5 )/(5) ,y= (15)/(5) \bigg \rgroup \\ \\ \\`

`\tiny: \implies (x,y) = \bigg \lgroup x = 1,y= 3 \bigg \rgroup \\ \\`