5. Find the ratio in which point P(k, 7) divides the segment joining A(8, 9) and B(1, 2). Also find k .

Let P (x, y) , A (x₁, y₁) and B (x₂, y₂) and be the given points.
Here, x = k, y = 7, x₁ = 8, y₁ = 9, x₂ = 1, y₂ = 2,
By Section formula y = (my₂+ny₁)/(m+n)
7 = m×2+n×9/(m+n)
7 = 2m+9n/(m+n)
7m+7n = 2m+9n
7m-2n = 9n-7n
5m = 2n
m/n = 2/5
Hence m:n = 2:5.
By Section formula, x = (mx₂+nx₁)/(m+n)
k = (2×1+5×8)/2+5
k = (2+40)/7
k = 42/7 = 6
Hence value of k is 6.
Hence P(6,7) divides the segment in the ration 2:5.