Q) Find the ratio in which line y = x divides the line segment joining the points (6, -3) and (1, 6).

Ans:

Let’s consider that the line y = x divided the line PQ in the ratio of m : n.

By section formula, if a point (x, y) divides the line joining the points (x₁, y₁​) and (x₂​, y₂​) in the ratio m : n, then coordinates of point R (x, y) =
(, )

Here, P (1, 6) = (x₁, y₁​)

Q (6, -3) = (x₂​, y₂​)

Since the line PQ is divided in the ratio of m : n, Hence the co-ordinates of point P:

x = =

Similarly, y = =

Since it is given that the point R (x,y) lies on the line y = x,

therefore we can transfer values of x & y in this line’s equation.

∴

∴ (- 3 m + 6 n)  = (6 m + n)

∴ 9 m = 5 n

∴ m : n = 5 : 9

Therefore, the line y = x divides the given line segment in the ratio of 5: 9.

We can check this by a diagram:

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