Question

If `ax^2+2hxy+by^2=0` , show that `(d^2y)/(dx^2)=0`

Answer 1

`ax^2+2hxy+by^2=0 .........(1)`

Differentiate w.r.t. x

`2ax+2hxdy/dx+2hy+2bydy/dx=0`

`therefore ax+hxdy/dx+bydy/dx+hy=0`

`(hx+by)by/dx=-1(ax+hy)`

`therefore dy/dx=-(ax+hy)/(hx+by) ..........................(2)`

From (1),we have

`ax^2+hxy+hxy+by^2=0`

`x(ax+hy)+y(hx+by)=0`

`x(ax+hy)=-y(hx+by)`

`-(ax+hy)/(hx+by)=y/x`

Put in (2),

`dy/dx=y/x`

Differentiate w.r.t. x

`(d^2y)/(dx^2)=(xdy/dx-y)/x^2`

`=(x(y/x)-y)/x^2`

`=0/x^2`

`=0`