the points which devides the line segment joining the points(-1,2)and (4,-5)internally in the ratio 2:3 is?

Answer 1

I wish I knew a simpler way to do this and I bet you do too!

First find the distance between the two points.
Let (x1, y1) = (-1, 2) and (x2, y2) = (4, -5).

Distance = sqrt[ (x2 - x1)^2 + (y2 - y1)^2 ]

= sqrt[ (4 - (-1))^2 + (-5 - 2)^2 ] = sqrt(74)

Now divide the line in the ratio 2:3.
(that is, 2 parts + 3 parts = length of line)

So, 2n + 3n = sqrt(74).
Therefore, 5n = sqrt(74) and n = sqrt(74) / 5.

n is 1 part, so 2 parts = 2n = 2*sqrt(74) / 5
and 3 parts = 3n = 3*sqrt(74) / 5.

Now the line can be divided in the ratio 2:3 in two
different ways, one where the dividing point is 2
parts away from one end, and the other where it
is 3 parts away from that same end.

Let's start with having the dividing point 2 parts away
from the point (-1, 2) and call the dividing point (x, y).

This distance is 2*sqrt(74) / 5 and the equation for
the distance is :
sqrt[ (x - (-1))^2 + (y - 2)^2 ] = 2*sqrt(74) / 5

Square both sides :
x^2 + 2x + 1 + y^2 - 4y + 4 = 4*74 / 25
or
(A) x^2 + y^2 + 2x - 4y + 5 = 4*74 / 25

For our second equation, we'll use the line that is
3 parts away from the point (4, -5). We know its
distance is 3*sqrt(74) / 5, and the equation is :
sqrt[ (x - 4)^2 + (y - (-5))^2 ] = 3*sqrt(74) / 5

Square both sides :
x^2 - 8x + 16 + y^2 + 10y + 25 = 9*74 / 25
or
(B) x^2 + y^2 - 8x + 10y + 41 = 9*74 / 25

Now do (B) - (A) :
-10x + 14y + 36 = 5*74 / 25 = 74 / 5

Multiply through by 5 / 2 to simplify :
-25x + 35y + 90 = 37
or, (C) 35y - 25x = -53

Now we need another equation to solve for x & y.
How about the equation to the line?
The slope is (y2 - y1) / (x2 - x1) = (-5 - 2) / (4 - (-1)) = -7/5.
If the equation to the line is y = mx + b, then we can plug
in the slope and one point, say (-1, 2), which we know is on
the line, to find b.
Thus, b = y - mx = 2 - (-7/5)(-1) = 2 - 7/5 = 3/5.
The equation to the line is therefore : y = (-7/5)x + 3/5, or
we can reformat it to (D) 5y + 7x = 3.
We now have the 2 simultaneous equations (C) and (D).
(C) 35y - 25x = -53
(D) 5y + 7x = 3
Solving gives : x = 1 and y = -4/5.
Thus, one of the dividing points is (1, -4/5).

I'm sure you can do the other one now.
Check if I'm right, that the other point is (2, -11/5).

Answer 2

Let position vector of required point be r
(-1 2) and (4 -5) are vectors.
r = (- 1 2) + (2/5).(5 -7)
r = (-1 2) + (2 - 14/5)
r = (1 - 4/5)
Point is (1, - 4/5)

Answer 3

Could you please explain what do you mean by dividing the line segment in the ratio 2:3???