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.

The given points are A(3, −5) and B(−4, 8). Let, x₁ = 3, y₁ = −5, x₂ = −4, y₂ = 8. Since AP/PB = K/1, the point P divides the line segment joining the points A and B in the ratio K:1. Using section formula:
(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)
Here m = K, n = 1
Coordinates of P = (K(−4) + 1(3))/(K + 1), (K(8) + 1(−5))/(K + 1) ⇒ (−4K + 3)/(K + 1), (8K − 5)/(K + 1)
It is given that, P lies on the line x + y = 0. Therefore,
(−4K + 3)/(K + 1) + (8K − 5)/(K + 1) = 0
⇒ −4K + 3 + 8K − 5 = 0
⇒ 4K − 2 = 0
⇒ 4K = 2
⇒ K = 1/2
So, the value of K is 1/2.