Q) In what ratio is the line segment joining the points (3, – 5) and (- 1, 6) divided by the line y = x ?

Ans: 

Method 1:

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

Step 2: Now if the given points A (3, -5) 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 = x, this point will satisfy the equation

∴

∴ 6 m – 5 n = 3 n – m

∴ 7 m = 8 n

∴ m : n = 8 : 7

Therefore, the line segment divides the line in ratio of 8:7.

Method 2: Let’s plot the points on the graph:

Let’s make an equation of the line passing through points A and B:
y – Y₁ = (x – X₁)
y – (- 5) = (x – 3)
y + 5 = (x – 3)
– 4 (y + 5) = 11 (x – 3)
– 4 y – 20 = 11 x – 33
– 2 y + 6 = x – 5
11 x + 4 y = 13

Since this line intersects the line x = y, hence for the intersection point, abscissa and ordinate values will be equal i. e. x = y
Hence, 11 x + 4 (x) = 13 or 15 x = 13 or x = ; Similarly, 11 (y) + 4 y = 13 or 15 y = 13 pr y = .
Let’s say this is point C.

Next, we calculate distance of C from A and B respectively. Hence AC = = = = =

Next, BC = = = = =

Let’s check the ratio of these lengths. Hence AC:BC = = = = = 8:7.

Therefore the ratio m:n = 8 :7

How to check your answer:

Here, Let’s consider that the ratio of m:n = 8:7. Hence, the coordinates of intersection point are: = = = .

Since this point lies on the line y = x, hence its abscissa and ordinate values should be equal. We can see that this is true for our coordinates of the intersection point, hence our answer is correct.

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