For the family of curves described by x^ 2 +xy+ky^ 2 =7

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

Problem

For the family of curves described by

x^{2}+xy+ky^{2}=7,

let

k

be a real constant. A particular curve in the family passes through the point

(1,2)

. (a) Determine the value of

k

that allows the point

(1,2)

to satisfy the given relation. (b) For general

k

, compute

\dfrac{dy}{dx}

in terms of

x

y

, and

k

. (c) Using your expression from part (b), find the value of

k

(if any) that makes the tangent line at

(1,2)

horizontal. (d) For the value of

k

obtained in part (c), write the equation of the tangent line at

(1,2)