Mathematics

Problem

For the curve given by

x^{2}+xy+y^{2}=3,

let

x=1.

a) Find all corresponding

y

–coordinates on the curve. b) For each point found in part (a), determine the slope of the curve. Provide the slopes in the same order as the points you list.

Intuitive explanation

This problem combines implicit differentiation with algebraic substitution. When you plug in $x=1$ , you'll get a quadratic equation in $y$ that yields two points—the curve passes through two places at that $x$ -value. Then, instead of solving for $y$ explicitly, you'll differentiate the entire equation implicitly with respect to $x$ to find $\frac{dy}{dx}$ , which gives you the slope at each point without needing a formula for $y(x)$ .

Intuitive explanation

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