Question

Find the ratio in which the point (-3, p) divides the line segment joining the points (-5, -4) and (-2, 3). Hence find the value of p.

Answer 1

(x1,y1)=(-5,-4), (x2,y2) = (-2,3)
(-3,p) is the point which divides joining the above points in the ratio k:1
by using section formula

[(k*(-2)+1*5)/(k+1) , (k*3+1*(-4))/(k+1)] = (-3,p)
equating x co ordinates
(-2k+5)/(k+1) = -3
-2k+5 = -3(k+1)
-2k+5 = -3k-3
-2k+3k=-3-5
k= -8---(1)

equating y co ordinates
(3k-4)/(k+1) = p
substitute k = -8
[3*(-8) -4]/(-8+1) = p

(-24-4)/-7 = p
-28/-7 =p
4=p
p=4

Answer 2

Let (-3,k) divide the line segment joining the points (-5,-4)and (-2,3) in the ratio m₁:m₂.Using the section formula , we get

(-3,k)=[(-2m₁-5m₂)/(m₁+m₂),(3m₁-4m₂)/(m₁+m₂)]

-3= (-2m₁-5m₂)/(m₁+m₂)

-3m₁-3m₂ =-2m₁-5m₂

m₁=2m₂ ,m₁/m₂=2:1

k = (3m₁-4m₂)/(m₁+m₂)

k =(6m₂-4m₂)/3m₂= 2m₂/3m₂=2/3