Question

(iii) Find the ratio in which the line segment joining the points A(3, 8) and
B(-9, 3) is divided by the Y-axis.
​

Answer 1

Given two points A(3, 8) and B(-9, 3). The line segment formed by joining these two points is divided by the y-axis. We have to find the ratio in which the line segment is divided.

Let the ratio in which the line segment is divided be k : 1 , also the coordinate of the point at which the y-axis intersects the line segment is (0, y) as on y-axis the x-coordinate is 0.

Now, using the section formula, the abscissa of the intersecting point is given by,

⇒ x = { 1(x₂) + k(x₁) } / (k + 1)

⇒ 0 = { 1(-9) + k(3) } / (k + 1)

⇒ 0(k + 1) = -9 + 3k

⇒ 3k - 9 = 0

⇒ 3k = 9

⇒ k = 3

Hence, The ratio in which the y-axis divides the given line segment is 1 : 3.

Some Information:

• The distance between two points (x₁, y₁) and (x₂, y₂) is given by:

⇒ D = √{ (x₂ - x₁)² + (y₂ - y₁)² }

• The coordinate of the midpoint of a line segment formed by two points (x₁, y₁) and (x₂, y₂) is given as,

⇒ C(X, Y) = { (x₁ + x₂)/2 , (y₁ + y₂)/2 }

Answer 2

Given :-

• The points A(3 , 8) and B(- 9 , 3) is divided by the Y-axis.

To Find :-

• What is the ratio in which the line segment is joined.

Formula Used :-

$\clubsuit$ Section Formula :

$\longmapsto \sf\boxed{\bold{\pink{(x , y) =\: \bigg(\dfrac{mx_2 + nx_1}{m + n} , \dfrac{my_2 + ny_1}{m + n}\bigg)}}}$

Solution :-

Let, the ratio in which the line AB is divided by the Y-axis be k : 1

Given :

• m = k
• x₂ = - 9
• n = 1
• x₁ = 3
• y₂ = 3
• y₁ = 8

$\implies \sf (0 , y) =\: \bigg(\dfrac{k(- 9) + 1(3)}{k + 1} , \dfrac{k+3 + 1(8)}{k + 1}$

$\implies \sf (0 , y) =\: \bigg(\dfrac{k \times (- 9) + 1 \times 3}{k + 1} , \dfrac{k \times 3 + 1 \times 8}{k + 1}\bigg)$

$\implies \sf (0 , y) =\: \bigg(\dfrac{- 9k + 3}{k + 1} , \dfrac{3k + 8}{k + 1}\bigg)$

Now,

$\implies \sf 0 =\: \dfrac{- 9k + 3}{k + 1}$

By doing cross multiplication we get,

$\implies \sf - 9k + 3 =\: 0$

$\implies \sf \cancel{-} 9k =\: \cancel{-} 3$

$\implies \sf 9k =\: 3$

$\implies \sf k =\: \dfrac{\cancel{3}}{\cancel{9}}$

$\implies \sf k =\: \dfrac{1}{3}$

$\implies \sf\bold{\red{k =\: 1 : 3}}$

$\therefore$ The ratio in which the line segment joining the points A(3 , 8) and B(- 9 , 3) is divided by the Y-axis is 1 : 3.