Question

A point P divides the line segment joining the points A (3, -5) and B (-4, 8) such that AP/PB=k/1. If P lies on the line x + y = 0, then find the value of k.

Answer 1

value of k=1/2

Solution : Let (x,y) be the coordinates of P.

As P divides the given line in k:1

m1=k m2=1

x1=3 x2=-4

y1=-5 y2=8

By section formula,

x = (m1x2+m2x1)/(m1+m2)

=>x= [k(-4)+1(3)]/(k+1)

=>x= (3-4k)/(k+1)

y = (m1y2+m2y1)/(m1+m2)

=>y = [k(8)+1(-5)]/(k+1)

=>y =(8k-5)/(k+1)

As point P lies on x+y=0

=>P[(3-4k)/(k+1) , (8k-5)/(k+1)] satisfy the

equation

=> (3-4k)/(k+1) +(8k-5)/(k+1)=0

=> 3-4k+8k-5 =0

=> 4k-2=0

=> k=1/2

Answer 2

k = 1/2

Step-by-step explanation:

Let the coordinates of P are (m,n).

Now, point P(m,n) divides the line AB, where A(3,-5) and B(-4,8), in the ratio of k : 1 internally, then the coordinates of P are given by

(m,n) ≡ [$\frac{3 \times 1 + (-4)\times k}{k + 1}, \frac{(-5) \times 1 + 8 \times k}{k + 1}$] ≡ [$\frac{3 - 4k}{k + 1}, \frac{8k - 5}{k + 1}$].

Now, this point P(m,n) lies on the straight line, x + y = 0

So, m + n = 0

⇒ $\frac{3 - 4k}{k + 1} + \frac{8k - 5}{k + 1} = 0$

⇒ 3 - 4k + 8k - 5 = 0

⇒ 4k = 2

⇒ k = 1/2 (Answer)