If ax² + 2hxy + by² = 1, then dy/dx equals

The correct option is: −(ax+hy)/hx+by

To find the derivative $\frac{dy}{dx}$ of the given implicit function, we perform implicit differentiation with respect to $x$ on both sides of the equation:

$a{x}^2+2hxy+b{y}^2=1$

Differentiating each term term-by-term with respect to $x$:

1. The derivative of $a{x}^2$ with respect to $x$ is:
$\frac{d}{dx}\left(a{x}^2\right)=2ax$

2. For the term $2hxy$, we use the product rule $\frac{d}{dx}\left(uv\right)=u\frac{dv}{dx}+v\frac{du}{dx}$:
$\frac{d}{dx}\left(2hxy\right)=2h[x\frac{dy}{dx}+y(1\left)\right]=2hx\frac{dy}{dx}+2hy$

3. For the term $b{y}^2$, we use the chain rule:
$\frac{d}{dx}\left(b{y}^2\right)=2by\frac{dy}{dx}$

4. The derivative of the constant on the right-hand side is:
$\frac{d}{dx}\left(1\right)=0$

Combining all the differentiated terms, we get:
$2ax+2hy+2hx\frac{dy}{dx}+2by\frac{dy}{dx}=0$

We can divide the entire equation by 2 to simplify it:
$ax+hy+hx\frac{dy}{dx}+by\frac{dy}{dx}=0$

Now, group the terms containing $\frac{dy}{dx}$ on one side and move the other terms to the right side:
$(hx+by)\frac{dy}{dx}=-(ax+hy)$

Solving for $\frac{dy}{dx}$, we divide both sides by $(hx+by)$:
$\frac{dy}{dx}=\frac{-(ax+hy)}{hx+by}$