The relation x^ 2 -xy+y^ 2 =3 represents a closed curve

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The relation x2−xy+y2=3x^{2}-xy+y^{2}=3 represents a closed curve. A point on this curve has x=1x=1 and lies in the first quadrant. (a) Determine the corresponding yy-coordinate of this point. (b) Differentiate the given relation to obtain dydx\dfrac{dy}{dx} in terms of xx and yy. (c) Evaluate dydx\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=1x=1 that lies in the fourth quadrant, and compare the two tangents.

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