Question

Example 9 : Find the ratio in which the y-axis divides the line segment joining the
points (5,- 6) and (-1,-4). Also find the point of intersection.​

Answer 1

let the coordinates of the point be (0,y) and the y axis divide the line segment in the ratio k:1

(0,y)={(-k+5)/k+1 , (-4k-6)/k+1}

=>0=-k+5/k+1

=>0=-k+5

=>k=5

y=-4k-6/k+1

=>y=-4(5)-6/5+1

=>y=-20-6/6

=>y=-26/6

thus the coordinates of the point is (0,-26/6)

Answer 2

Let the line segment A(5, -6) and B(-1, -4) is divided at point P(0, y) by y-axis in ratio m:n

:. x = $\frac{mx2+nx1}{m+n}$ and y = $\frac{my2+ny1}{m+n}$

Here, (x, y) = (0, y); (x1, y1) = (5, -6) and (x2, y2) = (-1, -4)

So , 0 = $\frac{m(-1)+n(5)}{m+n}$

=> 0 = -m + 5n

=> m= 5n

=> $\frac{m}{n}$ = $\frac{5}{1}$

=> m:n = 5:1

Hence, the ratio is 5:1 and the division is internal.Now,

y = $\frac{my2+ny1}{m+n}$

=> y = $\frac{5(-4)+1(-6)}{5+1}$

=> y = $\frac{-20-6}{6}$

=> y = $\frac{-26}{6}$

=> y = $\frac{-13}{3}$

Hence, the coordinates of the point of division is (0, -13/3).

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