If `(x-a)^2+(y-b)^2=c^2,` for some `c >0,` `p rov et h a t([1+((dy)/(dx))^2]^(3/2))/((d^2y)/(dx^2))i sacon s t a n tin d e p e n d e n tof` a and b.

Options

A. `-c`
B. `-(c) /(a)`
C. `- (a)/(c) `
D. `- abc `

Answer

A

Explanation

Given , ` (x - a)^(2) + (y - b)^(2) = c^(2)` …(i)
On differentiating both sides w.r.t.x, we get
` 2(x-a) + 2 (y - b)(dy)/(dx) = 0 `
`rArr (dy)/(dx) = - (x-a)/(y-b) ` …(ii)
Again , differentiating both sides w.r.t.x, we get
`(d^(2) y)/( dx^(2) )= - ((y - b) *1- (x -a)(dy)/(dx))/((y - b)^(2))`
` = - ((y - b) *1-(x -a) (-(x -a)/(y-b)))/((y - b)^(2))` [ from Eq. (ii) ]
`= - ((y-b)^(2) + (x -a)^(2))/((y- b)^(3)) = - (c^(3))/((y-b)^(3))` [ from Eq. (i) ]
Now , `([1 + ((dy)/(dx))^(2)]^(3//2))/((d^(2)y)/(dx^(2)))=([1 + ((x-a)/(y-b))^(2)]^(3//2))/((c^(2))/((y-b)^(3)))`
` ({(y-b)^(2) + (x - a)^(2)}^(3//2))/({(y-b)^(2)}^(3//2) ((-c^(2)))/((y-b)^(3)) )=((c^(2))^(3//2))/(-c^(2)) = (c^(3))/(-c^(2) ) = - c ` .