Find the tangent to the curve (x^4+y^4)^2=2013(x^2-y^2) at the origin.

Name: Problem 30, Chapter 8: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-03-03T13:26:48-08:00
Duration: 2 min 24 s
Channel: Charles Machakwa
Description: Find the tangent to the curve (x^4+y^4)^2=2013(x^2-y^2) at the origin.

step 1 Here we have we are to find the tangent to the given curve at the origin. Now differentiating im

Find the tangent to the curve (x^4+y^4)^2=2013(x^2-y^2) at...

Key Concepts

Implicit Differentiation

This technique involves differentiating both sides of an equation that defines a curve implicitly with respect to a variable (usually x) to find the slope (dy/dx) of the tangent at a given point. It is especially useful for curves that are not given in explicit form, and it plays a central role in analyzing the behavior of functions near specific points.

Singular Points

A singular point on a curve is one where the usual differentiability conditions fail—typically, partial derivatives vanish or do not yield a unique tangent direction. At such points, standard implicit differentiation may not apply directly, necessitating alternative methods to determine the curve’s local linear approximation.

Tangent Cone Method

When analyzing a singular point of an algebraic curve, one often examines the lowest order (or homogeneous) terms in the polynomial expansion. The equation formed by these terms, known as the tangent cone, provides the potential tangent directions at that point by indicating the leading-order behavior of the curve near the singularity.

Substitution Technique (y = mx)

A common approach to finding tangent lines at a singular point is to substitute a linear relation, such as y = mx, into the curve’s equation. This substitution reduces the problem to finding the values of m that satisfy the leading-order terms. The resulting equation, solved for m, reveals the slopes of the tangent lines, offering insight into the curve’s local structure.

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