Mathematics

Problem

The relation

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

represents a closed curve. A point on this curve has

x=1

and lies in the first quadrant. (a) Determine the corresponding

y

-coordinate of this point. (b) Differentiate the given relation to obtain

\dfrac{dy}{dx}

in terms of

x

and

y

. (c) Evaluate

\dfrac{dy}{dx}

at the first-quadrant point found in part (a). State whether the tangent line there is increasing, decreasing, horizontal, or vertical. (d) Repeat parts (a)–(c) for the point on the curve with

x=1

that lies in the fourth quadrant, and compare the two tangents.

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