If ax^2+2hxy+by^2=3, then frac{d^2y}{dx^2}= | vedclass

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(C) Given the equation: $ax^2+2hxy+by^2=3$. Differentiating with respect to $x$: $2ax + 2h(y + x\frac{dy}{dx}) + 2by\frac{dy}{dx} = 0$. Dividing by $2

Vedclass · Accepted Answer

$\frac{3(h^2-ab)}{(hx+by)^3}$

Vedclass · Answer

(C) Given the equation: $ax^2+2hxy+by^2=3$. Differentiating with respect to $x$: $2ax + 2h(y + x\frac{dy}{dx}) + 2by\frac{dy}{dx} = 0$. Dividing by $2$: $ax + hy + hx\frac{dy}{dx} + by\frac{dy}{dx} = 0$. Thus, $\frac{dy}{dx} = -\frac{ax+hy}{hx+by}$. Differentiating again with respect to $x$: $\frac{d^2y}{dx^2} = -\frac{(hx+by)(a+h\frac{dy}{dx}) - (ax+hy)(h+b\frac{dy}{dx})}{(hx+by)^2}$. Substituting $\frac{dy}{dx} = -\frac{ax+hy}{hx+by}$ and simplifying using the original equation $ax^2+2hxy+by^2=3$, we get: $\frac{d^2y}{dx^2} = \frac{3(h^2-ab)}{(hx+by)^3}$.

If $ax^2+2hxy+by^2=3$, then $\frac{d^2y}{dx^2}=$

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