If the distance between the points (7, 1, -3) and (4, 5, λ) is 13 units, then what is one of the values of λ?  

Concept:

Let A = (x,y,z) and B = (a, b, c) be any two points then distance between them is ,

By distance formula: \(\rm AB = \sqrt {(x-a)^2+(y-b)^2+(z-c)^2}\)

Calculations:

 Consider A = (7, 1, -3)  = (x, y, z)

and B = (4, 5 , \(λ\)) = (a, b, c)

given the distance between the points A = (7, 1, -3) and  B= (4, 5, λ) is 13 units.

By distance formula

\(\rm AB = \sqrt {(x-a)^2+(y-b)^2+(z-c)^2}\)

⇒\(\rm 13 = \sqrt {(7-4)^2+(1-5)^2+(-3-λ)^2}\) 

Squaring on both side 

⇒\(\rm 13^2 = {(7-4)^2+(1-5)^2+(-3-λ)^2}\)

⇒169 = 9 + 16 + 9 + 6\(λ\) + \(λ^2\)

⇒ \(λ^2\)+ 6\(λ\) - 135 = 0

⇒\(\rm (λ+15)(λ-9) = 0\)

⇒ \(\rmλ = -15, 9\)