In what ratio is the line segment joining a (- 2 3 and B 3 7 is divided by the Y axis also find the coordinates of the point of intersection of AB and the y axis?

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i In what ratio is the line segment joining the points 2, 3 and 3, 7 divided by the y axis ? Also, find the coordinates of the point of division. ii In what ratio is the line segment joining 3, 1 and 8, 9 divided at the point 5, 21/5 ?

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Let AB be divided by the x-axis in the ratio :1 k at the point P.

Then, by section formula the coordination of P are

`p = ((3k-2)/(k+1) , (7k-3)/(k+1))`

But P lies on the y-axis; so, its abscissa is 0.
Therefore , `(3k-2)/(k+1) = 0`

`⇒ 3k-2 = 0 ⇒3k=2 ⇒ k = 2/3 ⇒ k = 2/3 `

Therefore, the required ratio is `2/3:1`which is same as 2 : 3
Thus, the x-axis divides the line AB in the ratio 2 : 3 at the point P.

Applying `k= 2/3,`  we get the coordinates of point.

`p (0,(7k-3)/(k+1))`

`= p(0, (7xx2/3-3)/(2/3+1))`

`= p(0, ((14-9)/3)/((2+3)/3))`

`= p (0,5/5)`

= p(0,1)

Hence, the point of intersection of AB and the x-axis is P (0,1).