Distance of point P on the curve `y=x^(3//2)` which is nearest to the point M (4, 0) from origin is

Options

A. `sqrt((112)/(27))`
B. `sqrt((100)/(27))`
C. `sqrt((101)/(9))`
D. `sqrt((112)/(9))`

Answer

A

Explanation

AP will be least if AP is normal at point P.
`therefore" Slope of AP" xx"Slope of tangent at point P"=-1`
Slope of tangent at point `P=((dy)/(dx))_(P)=(3)/(2)sqrtx_(1)`
`therefore" "(3)/(2)sqrtx_(1).(x_(1)^(3//2)-0)/(x_(1)-4)=-1`
`rArr" "3x_(1)^(2)=8-2x_(1)`
`rArr" "3x_(1)^(2)+2x_(1)-8=0`
`rArr" "(x_(1)+2)(3x_(1)-4)=0`
`rArr" "x_(1)=(4)/(3)`
`rArr" Point P"((4)/(3),((4)/(3))^(3//2))`
`rArr" OP"=sqrt((112)/(27))`