Question

Fnvelope of the family of lines $x=m y+1 / m$ is $\ldots$

   Fnvelope of the family of lines $x=m y+1 / m$ is $\ldots$

Higher Engineering Mathematics

Higher Engineering Mathematics

Grewal B.S. 42nd Edition

Chapter 4, Problem 15

Instant Answer

Solved by Expert Ankit Gupta

07/01/2020

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We have the family of lines given by $x = my + \frac{1}{m}$. Show more…

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Fnvelope of the family of lines $x=m y+1 / m$ is $\ldots$

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Key Concepts

Envelope of a Family of Curves

This concept refers to a curve that is tangent to every member of a family of curves. In other words, the envelope is the boundary that 'envelops' all curves in the family. Finding the envelope typically involves determining the condition under which a point lies on a member of the family and is the point of tangency, meaning it satisfies both the original family’s equation and an accompanying differentiability condition with respect to the family parameter.

Parameter Elimination via Differentiation

A key technique in determining an envelope involves introducing a parameter that defines the family of curves and then eliminating it. This is often done by differentiating the equation of the family with respect to the parameter and solving the resulting equation together with the original family equation to remove the parameter. The resulting relationship between the variables represents the envelope of the family.

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