Question

if point t divides the segment ab with a (-7,4) and b (-6,-5)in the ratio 7:2 find the co-ordinate of the.

Answer 1

Lets call the point (-3,5 ) ‘A’.

Lets call the point (4,2) ‘B’.

Let the line AB intersect the y-axis at ‘C’.

Now we need to calculate the distances ‘AC’ and ‘CB’.

For that we need to find the coordinates of the point ‘C’.

The slope of ‘AB’ is (y2 - y1) / (x2 - x1) = (2–5) / (4- (-3) ) = -3 / 7.

The equation of a line can be represented as

y = mx + c

y = (-3/7)x + c. Lets put (4,2) in this equation.

(Since (4,2) lies on the line, it must satisfy the equation of the line.

=> 2 = -12/7 + c OR c = 26/7.

Hence the equation of the line is

y = (-3/7)x + 26/7.

OR 7y + 3x = 26. ……………. (1)

Check this for (4,2) .

LHS : 14 + 12 = 26 = RHS

Check this for (-3, 5)

LHS : 35 - 9 = 26 = RHS.

Hence the equation is correct.

Also, if we put x=0 in the equation (1) above, we can find the point of intersection of the line with the y-axis.

When we do that, we get y = 26/7.

Hence, A : (-3,5) ;; B : (4,2) ;; C : (0,26/7)

The distance between two points can be found out by the distance formula

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

AB = sqrt [(7^2) + (-3)^2 ] = 7.615

AC = sqrt [(3)^2 + (5–26/7)^2 ] = 3.264

Hence, CB = 7.615 - 3.264 = 4.351

AC/AB = 3.264/7.615 = 0.4286

CB/AB = 1–0.4286 = 0.5714

OR, the y-axis divides the given line in the ratio of 42.86 : 57.14 of their lengths.