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Magazine Chapter 12: Problem 31
Prove that there is exactly one line of a given slope \(m\) that is tangent to the parabola \(x^{2}=4 p y,\) and show that its equation is \(y=m x-p m^{2}\)
Short Answer
Expert verified
The line tangent to \(x^2=4py\) with slope \(m\) is unique and given by \(y=mx-pm^2\).
Step by step solution
01
Set Up the Problem
To find the line tangent to the parabola \(x^2 = 4py\) with slope \(m\), we'll use the derivative of the parabola to express the slope of the tangent line as a function of \(x\). Recall that for a parabola \(y = \frac{x^2}{4p}\), the derivative \(\frac{dy}{dx}\) gives the slope of the tangent.
02
Differentiate to Find Slope
Differentiate the equation \(x^2 = 4py\) implicitly: \(\frac{d}{dx}(x^2) = \frac{d}{dx}(4py)\) which results in \(2x = 4p \frac{dy}{dx}\). Solving for \(\frac{dy}{dx}\), the slope of the tangent line is \(\frac{dy}{dx} = \frac{x}{2p}\). Set this equal to the given slope \(m\), thus \(\frac{x}{2p} = m\). Solve for \(x\): \(x = 2pm\).
03
Find Corresponding y-value
Substitute \(x = 2pm\) back into the equation \(x^2 = 4py\) to find the y-coordinate of the tangent point. \((2pm)^2 = 4py\) simplifies to \(4p^2m^2 = 4py\). Solving for \(y\) results in \(y = pm^2\).
04
Equation of the Tangent Line
We now have a point \((2pm, pm^2)\) on the parabola. Using the point-slope form of a line, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is \((2pm, pm^2)\), the equation becomes: \(y - pm^2 = m(x - 2pm)\). Simplifying this, we get \(y = mx - 2pm^2 + pm^2\), which results in \(y = mx - pm^2\).
05
Verify Uniqueness
To prove there's exactly one such line, consider that each tangent line's slope \(m\) derives a unique \(x = 2pm\), which corresponds to a unique \(y = pm^2\). Since these determine the line's point, no different points on the parabola can give the same slope \(m\) without leading back to the same point.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
A parabola is a specific type of curve seen in mathematics, often represented in a quadratic form. The standard equation of a simple parabola is given by \(x^2 = 4py\), where \(p\) is a constant that determines the "width" and direction of the parabola. This curve is symmetrical, and its general shape resembles a "U" or an inverted "U" depending on its orientation.
- It has a vertex, which is the highest or lowest point on the curve.
- The axis of symmetry is a vertical line that passes through the vertex.
- Understanding a parabola's orientation and vertex is key to determining other properties, like the focus and directrix, which also help in defining its characteristics.
Slope
Slope is a measure of the steepness or angle of a line. In the context of a tangent line to a curve, the slope describes how fast the curve rises or falls at a particular point. It is defined as the ratio of the vertical change to the horizontal change between two points on the line.
- For a given line in the plane, the slope \(m\) can be calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
- In the case of a tangent line, the slope refers to the instantaneous rate of change at a specific point.
- Determining the slope of a tangent line is often done using derivatives, which provides a precise measure at each point on a curve.
Derivative
The derivative is a fundamental concept in calculus that represents the rate of change of a function with respect to a variable. It is symbolized as \(\frac{dy}{dx}\) for a function \(y = f(x)\).
- For the parabola \(y = \frac{x^2}{4p}\), the derivative provides the slope of the tangent line at any point \(x\).
- Calculating the derivative involves rules of differentiation, such as the power rule, product rule, or implicit differentiation, depending on the form of the function.
- In the exercise at hand, the derivative of \(x^2 = 4py\) with respect to \(x\) results in \(\frac{dy}{dx} = \frac{x}{2p}\), which is set equal to the desired slope \(m\) to find the tangent point.
Point-Slope Form
The point-slope form is a way of writing the equation of a line given its slope and a point on the line. It is particularly useful for finding the equation of a tangent line. The formula is expressed as:
\[y - y_1 = m(x - x_1)\]
where \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope.
\[y - y_1 = m(x - x_1)\]
where \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope.
- This form is versatile and simplifies the process of writing the equation of a line when the slope and a point are known.
- For example, in deriving the tangent line to the parabola \(x^2 = 4py\), we use the point \((2pm, pm^2)\) obtained from differentiating and manipulating our initial parabola equation.
- Substitution into the point-slope form gives the equation of the tangent line \(y = mx - pm^2\), confirming it is tangent to the parabola with the slope \(m\).
