Let `x_(0), y_(0)` be fixed real numbers such that `x_(0)^(2)+y_(0)^(2)gt1`. If x, y are arbitrary real numbers such that `x^(2)+y^(2)le1`, then the minimum value of `(x-x_(0))^(2)+(y-y_(0))^(2)` is

Options

A. `(sqrt(x_(0)^(2))+(y_(0)^(2))-1)^(2)`
B. `x_(0)^(2)+y_(0)^(2)-1`
C. `(|X_(0)|+|Y_(0)|-1)^(2)`
D. `(|X_(0)|+|Y_(0)|)^(2)-1`

Answer

A

Explanation

`x_(0)^(2)+y_(0)^(2)gt1" "x_(0)-y_(0)" fixed"`
`x, y" "aribitary"`
`x^(2)+y^(2)li1" , Let "x=costheta`
`" "y=sintheta`
for men
`z=(x-x_(0))^(2)+(y-y_(0))^(2)`
`z=x^(2)+x_(0)^(2)+y^(2)+y_(0)^(2)-2(x" "x_(0)+y" "y_(0))`
`" Put "x-costheta,y=sin theta`
`z=x_(0)^(2)+y_(0)^(2)-2(x_(0)" "costheta+y_(0)" " sintheta)`
`(dz)/(dtheta)rArr0-2(-x_(0)sin theta+y_(0) cos theta)`
`(dz)/(dtheta)=0`
`-x_(0) sin theta= -y_(0)cos theta`

`tan theta=(y_(0))/(x_(0))`
` sin theta =(y_(0))/(sqrt(x_(0)^(2)+y_(0)^(2)))`
`costheta(x_(0))/(sqrt(x_(0)^(2)+y_(0)^(2)))`
` x=(x_(0))/(sqrt(x_(0)^(2)+y_(0)^(2))),y=(y_(0))/(sqrt(x_(0)^(2)+y_(0)^(2)))`
`z=((x_(0))/sqrt(x_(0)^(2)+y_(0)^(2))-x_(0))^(2)+((y_(0))/sqrt(x_(0)^(2)+y_(0)^(2))-y_(0))^(2)`
`x_(0)2((1)/(sqrt(x_(0)^(2)+y_(0)^(2)))-1)^(2)+y_(0)2((1)/(sqrt(x_(0)^(2)+y_(0)^(2)))-1)^(2)`
`(x_(0)^(2)+y_(0)^(2))((1-sqrt(x_(0)^(2)+y_(0)^(2))))/((sqrt(x_(0)^(2)+y_(0)^(2)))^(2))`
`(1-sqrt(x_(0)^(2)+y_(0)^(2)))^(2)rArr(sqrt(x_(0)^(2)+y_(0)^(2))-1)^(2)`