Question

If (a, b) is the mid-point of the line segment joining the points A(10, –6) and B(k, 4) and a – 2b = 18, find the value of k and the distance AB.

Answer 1

Since, (a, b) is the mid-point of line segment AB.

∴ (a, b) = `((10 + "k")/2, (-6 + 4)/2)`   ...`["Since, mid-point of a line segment having points"  (x_1, y_1)  "and" (x_2, y_2) = ((x_1 + x_2)/2, (y_1 + y_2)/2)]`

⇒ (a, b) = `((10 + "k")/2, -1)`

Now, equating coordinates on both sides, we get

∴ a = `(10 + "k")/2`  ...(i)

And b = –1   ...(ii)

Given, a – 2b = 18

From equation (ii),

a – 2(–1) = 18

⇒ a + 2 = 18

⇒ a = 16

From equation (i),

16 = `(10 + "k")/2`

⇒ 32 = 10 + k

⇒ k = 22

Hence, the required value of k is 22.

⇒ k = 22

∴ A = (10 – 6), B = (22, 4)

Now, distance between A(10, –6) and B(22, 4),

AB = `sqrt((22 - 10)^2 + (4 + 6)^2`   ...`[∵ "Distance  between the point"  (x_1, y_1)  "and"  (x_2, y_2), d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)]`

= `sqrt((12)^2 + (10)^2`

= `sqrt(144 + 100)`

= `sqrt(244)`

= `2sqrt(61)`

Hence, the required distance of AB is `2sqrt(61)`.