A particle moves in the plane along a path described implicitly by x^ 2 -xy+3y^ 2 =27

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

A particle moves in the plane along a path described implicitly by

x^{2}-xy+3y^{2}=27,

where

x

and

y

are measured in meters. At time

t_{0}

the particle is known to be at a point on the path with

y=2

and

x>0

. (a) Find all

x

-coordinates on the path that correspond to

y=2

. Show how you obtain each value. (b) For each point found in part (a), calculate the slope of the tangent line to the path. (c) Choose the point from part (b) whose tangent slope is positive. At that point, write the equation of the normal line in the form

Ax+By+C=0

with integer coefficients satisfying

\gcd(A,B,C)=1

. Justify the arithmetic used to obtain your final coefficients. (d) At the point selected in part (c), the particle’s instantaneous speed is

5\text{ m/s}

and its velocity vector is tangent to the path. Express the velocity vector in component form

\langle v_{x},v_{y}\rangle

and verify that its magnitude is

5

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