Question

the shortest distance between the point (3/2,0) and the curve y=
$\sqrt{x}$
is​

Answer 1

Answer:

the shortest distance between the point (3/2,0) and the curve y=  √x ,

is​ 1.118

Step-by-step explanation:

Let say point at curve y = √x which is at shortest distance

= x₁ , y₁

y₁ = √x₁

Let say Distance= D

D² = (x₁ - 3/2)²  + (y₁ - 0)²

=> D² = x₁² + 9/4 - 3x₁ + (√x₁)²

=> D² = x₁² + 9/4 - 3x₁ + x₁

=> D² = x₁² + 9/4 - 2x₁

=> D = (x₁² + 9/4 - 2x₁)^(1/2)

now we need to get minimum value of D

lets differentiate

d D/ d x  = (2x - 2) . 1 / (2(x₁² + 9/4 - 2x₁)^(1/2))

2x -2 = 0

=> x = 1

y = 1 or - 1

(1 , 1) or (1, -1) is closet point

Distance² = (1 -3/2)² + (1-0)²   or (1 -3/2)² + (-1-0)²

=> D² = 1/4 + 1

=> D = √5 / 2

=>D = 2.236/2

=> D = 1.118