Q) Find the ratio in which the line segment joining the points (5, 3) and (–1, 6) is divided by Y-axis.

Ans: 

Method 1:

Step 1:  By section formula, coordinates of point P (X, Y) which lies between two points (x₁, y₁), (x₂, y₂) will be given by:

P (X,Y) =

here, the point divides the line in the ratio of m_{1 :} m₂

Step 2:  Now if the given points A (5, 3) and B(- 1, 6) are divided in the ratio of m : n, then:

coordinates of dividing point (x, y) =

=

Step 3:  Next, since this point lies on line Y axis, it means x = 0, this point will satisfy the equation

∴ = 0

∴ – m + 5 n = 0

∴ m = 5 n

∴ m : n = 5 : 1

Therefore, the line segment divides the line in ratio of 5:1.

Method 2: if we plot the given points and connect them:

Let’s make an equation of the line passing through points A and B:
y – Y₁ = (x – X₁)
y – 3 = (x – 5)
y – 3 = (x – 5)
y – 3 = (x – 5)
– 2 (y – 3) = (x – 5)
– 2 y + 6 = x – 5
x + 2y – 11 = 0

Since, at Y-axis, x = 0, ∴ y = = 5.5
Hence, the line x + 2y – 11 = 0, intersects Y -axis at (0, 5.5). 
Let’s say this is point C.

Now, Distance between point A (5,3) and C (0, 5.5)
=
=
=

Now, Distance between point B (-1,6) and C (0, 5.5)
=
=
=

Now, Ratio between AC/ BC =
=
=
=
=
=

Therefore, m:n = 5:1

How to check your answer:

Here, Let’s consider that the ratio of m:n = 5:1. Hence, the coordinates of the intersection point are: = = = = (0, 5.5)

Since this point lies on the line y – axis, where x = 0. We can see that this is true for our coordinates of the intersection point, hence our answer is correct.

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