A one-parameter family of circles in the plane can be described implicitly by

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

Problem

x^{2} + y^{2} + C_1 x + C_2 y = 0

, where

C_1

and

C_2

are real constants. (1) Differentiate the implicit relation with respect to

x

to obtain an equation involving

x

y

\dfrac{dy}{dx}

C_1

, and

C_2

. (2) Eliminate the constants

C_1

and

C_2

to derive a first-order differential equation in

x

y

, and

\dfrac{dy}{dx}

that is satisfied by every circle in the family. Give your answer in explicit differential form. (3) Interpret geometrically the differential equation obtained in part (b) by describing the relationship it imposes between the slope of the tangent line to a circle from the family at

(x, y)

and the position vector

(x, y)

Related theory

Functions
Advanced Differentiation
Implicit and Parametric Differentiation
Derivatives of implicit functions
Continuity and Differentiability

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