Question

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

Short answer

The envelope is the parabola \( y = \frac{x^2}{4} \).

Step-by-step solution

Understand the Family of Lines

The given family of lines is expressed as \( x = m y + \frac{1}{m} \), where \( m \) is a parameter. As \( m \) changes, the line changes because both the slope \( m \) and the intercept \( \frac{1}{m} \) depend on it. We aim to find a curve that is tangent to each member of this family at some point, known as the envelope of the family.

Differentiate with Respect to the Parameter

Differentiate the equation of the line \( x = m y + \frac{1}{m} \) implicitly with respect to the parameter \( m \). This gives:\[ \frac{dx}{dm} = y - \frac{1}{m^2} \]

Set Differential Equation to Zero

To find the envelope, set the derivative from Step 2 equal to zero, since the envelope must be tangent to the lines, meaning no change with respect to \( m \):\[ y - \frac{1}{m^2} = 0 \]

Solve for y in Terms of m

Solve the equation obtained in Step 3 for \( y \) in terms of \( m \):\[ y = \frac{1}{m^2} \]

Substitute y into the Original Equation

Substitute \( y = \frac{1}{m^2} \) from Step 4 back into the original equation \( x = m y + \frac{1}{m} \):\[ x = m \left( \frac{1}{m^2} \right) + \frac{1}{m} = \frac{1}{m} + \frac{1}{m} = \frac{2}{m} \]

Eliminate the Parameter

From Steps 4 and 5, you have two equations:1. \( y = \frac{1}{m^2} \)2. \( x = \frac{2}{m} \)Eliminate \( m \) to find the relationship between \( x \) and \( y \), use \( m = \frac{2}{x} \) and substitute into the equation for \( y \):\[ y = \frac{1}{\left( \frac{2}{x} \right)^2 } = \frac{x^2}{4} \]

Write the Equation of the Envelope

The envelope of the family of lines is the curve described by the equation:\[ y = \frac{x^2}{4} \]

Key concepts

Differential Calculus

Differential calculus is a branch of mathematics that deals with the study of how things change. It focuses on the concept of the derivative, which measures how a function changes as its input changes. In simpler terms, it is used to find the rate of change or slope of a curve at any given point.

In the context of finding the envelope of a family of lines, differential calculus helps us understand how the position of a line varies as its defining parameter shifts.

When dealing with a family of lines, like in the exercise where lines are defined as \( x = m y + \frac{1}{m} \), differential calculus lets us explore how each line changes as the parameter \( m \) changes.

This aids in determining points that remain stationary, namely those that belong to the envelope of the family.

Tangency Condition

Finding the envelope involves a tangency condition, which means that the envelope we seek is tangent to each line of the family at some point. Tangency here implies that for a particular value of the parameter, the line and the envelope share a common point, and their slopes are equal at that point.

This concept is core to developing the envelope because

• The envelope "envelops" or "touches" all lines in the family without crossing them.
• It serves as the limit or boundary beyond which the lines do not extend.

In the exercise, by using the derivative \( \frac{dx}{dm} = y - \frac{1}{m^2} \) and setting it to zero, we set the stage for ensuring tangency. This is because the zero derivative indicates no change with respect to the parameter, a necessary condition for tangency.

Implicit Differentiation

Implicit differentiation is a powerful technique used when you cannot easily express one variable explicitly in terms of another. It allows us to find derivatives even when variables are intermixed in an equation.

In the given exercise, we are using implicit differentiation to work with the equation \( x = m y + \frac{1}{m} \), which is not solved for one variable in terms of another.

By differentiating implicitly,

• The relationship between \( x \) and \( m \) is maintained.
• We can quantify the rate at which \( x \) changes with \( m \) without isolating \( x \).

This step is crucial as it leads to the differential equation \( y - \frac{1}{m^2} = 0 \), a stepping stone to identifying the envelope by ensuring tangency conditions are met.

Parameter Elimination

Parameter elimination involves removing the parameter from a set of equations to find a direct relationship between the remaining variables. It's like simplifying a story by removing an unnecessary character that doesn't need to be there at the end.

In the exercise, we have equations involving \( m \):

• \( y = \frac{1}{m^2} \)
• \( x = \frac{2}{m} \)

By expressing \( m \) in terms of \( x \) \( (m = \frac{2}{x}) \), we eliminate \( m \) by substituting it back into \( y \).

Through this process, we derive the clean relationship \( y = \frac{x^2}{4} \) that represents the envelope. Parameter elimination allows us to seamlessly move from a complex set of relationships involving a parameter to a straightforward equation of the desired curve.