If `a x^2+2h x y+b y^2=1,t h e n(d^(2y))/(dx^2)` is `(h^2-a b)/((h x+b y)^2)` (b) `(a b-h^2)/((h x+b y)^2)` `(h^2+a b)/((h x+b y)^2)` (d) none of these

Options

A. `(h^(2)-ab)/((hx+by)^(3))`
B. `(ab-h^(2))/(hx+by)^(2)`
C. `(h^(2)+ab)/(hx+by)^(2)`
D. none of these

Explanation

`ax^(2)+2hxy+by^(2)=1`
Differentiating both sides w.r.t. x, we get
`2ax+2hx(dy)/(dx)+2hy+2by(dy)/(dx)=0`
`"or "(dy)/(dx)=-(ax+hy)/(hx+by)`
Again differentiating w.r.t. x, we get
`(d^(2)y)/(dx^(2))=-[((hx+by)(a+h(dy)/(dx))-(ax+hy)(h+b(dy)/(dx)))/((hx+by)^(2))]`
`=-([y(ab-h^(2))+(dy)/(dx)(h^(2)x-abx)])/((hx+by)^(2))`
`=((h^(2)-ab)(y-x(dy)/(dx)))/((hx+by)^(2))`
`=((h^(2)-ab))/((hx+by)^(2))[y+x(ax+hy)/(hx+by)]`
`(h^(2)-ab)/((hx+by)^(3))`