The line x-y=2 cuts the join of (3,-1) and (8,9) in the ratio of ?

Answer 1

First, find the coordinates of the point where the line x-y=2 intersects the line passing through (3,-1) and (8,9).

Let the point of intersection be (x,y). Since the line passes through both points, it satisfies the equation of the line passing through the two points:

(y-(-1))/(x-3) = (9-(-1))/(8-3)
(y+1)/(x-3) = 10/5
(y+1)/(x-3) = 2
y+1 = 2x - 6
y = 2x - 7

Since the point (x,y) also lies on the line x-y=2, we have:
x - (2x - 7) = 2
x - 2x + 7 = 2
-x + 7 = 2
x = 5

Substitute x=5 into y=2x-7:
y = 2*5 - 7
y = 10 - 7
y = 3

So, the point of intersection of the lines x-y=2 and the line passing through (3,-1) and (8,9) is (5,3).

Now, calculate the ratio in which this point divides the line joining (3,-1) and (8,9):

Let the ratio be k:1.

Using the section formula:
x = (k*x2 + x1)/(k+1)
5 = (k*8 + 3)/(k+1)

5(k+1) = 8k + 3
5k + 5 = 8k + 3
5 = 3k + 3
2 = 3k
k = 2/3

Therefore, the line x-y=2 cuts the join of (3,-1) and (8,9) in the ratio of 2:3.