Question

Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–1, –4). Also find the coordinates of the point of division.

Answer 1

Let (0, α) be a point on the y-axis dividing the line segment AB in the ratio k : 1.

Now, using the section formula, we get

`(0,alpha)=((-"k"+5)/("k"+1),(-4"k"-6)/("k"+1))`

`=>(-"k"+5)/("k"+1)=0,(-4"k"-6)/("k"+1)=alpha`

Now, 

`(-"k"+5)/("k"+1)=0`

⇒ −k + 5 = 0

⇒ k = 5

Also,

`(-4"k"-6)/("k"+1)=alpha`

`=>(-4xx5-6)/(5 +1)=alpha`

`=>alpha=(-26)/6`

`=>alpha=-13/3`

Thus, the y-axis divides the line segment in the ratio k : 1, i.e. 5 : 1.

Also, the coordinates of the point of division are (0, α), i.e `(0,-13/3)`

Answer 2

The ratio in which the y-axis divides two points (x₁ , y₁)  and  (x₂ , y₂)  is  \[\lambda: 1\] 

The co-ordinates of the point dividing two points (x₁ , y₁)  and (x₂ , y₂)   in the ratio m : n  is given as,

`(x , y) = ((lambdax_2 + x_1)/(lambda + 1 )) ,((lambda"y"_2 + "y"_1)/(lamda + 1))`  where, `lambda = "m"/"n"`

Here the two given points are A(5,−6) and B(−1,−4).

`(x, "y") = ((-lambda + 5)/(lambda + 1),(- 4lambda - 6)/(lambda + 1))`

Since, the y-axis divided the given line, so the x coordinate will be 0.

\[\frac{- \lambda + 5}{\lambda + 1} = 0\]

\[\lambda = \frac{5}{1}\]

Thus the given points are divided by the y-axis in the ratio  5 : 1.

The co-ordinates of this point (x, y) can be found by using the earlier mentioned formula.

`(x , "y" ) = ((5/1 (-1) + (5) )/(5/1 + 1)) , ((5/1(-4)+(-6))/(5/1 +1))`

`(x , "y") = (0/6) , (-26/6)`

`(x , "y")  = ( 0 , - 26/6)`

Thus the co-ordinates of the point which divides the given points in the required ratio are `(0,-26/6)`.