Problem 56 - Calculus Single Variable

Question

Suppose that \(A \neq 0 .\) The locus of $$ x^{2 / 3}+y^{2 / 3}=a^{2 / 3} $$ is called an astroid \(\mathcal{A}_{a} .\) Let \(T\) be a tangent line to the astroid \(\mathcal{A}_{a} .\) Find a formula for the distance between the two intercepts of \(T .\) How does this distance depend on \(T ?\)

Step-by-Step Solution

Answer

The distance is \(a|\sec(t) + \csc(t)|\), depending on \(t\) through \(\sec(t)\) and \(\csc(t)\).

Step 1: Parametrize the Astroid

The astroid given by \(x^{2/3} + y^{2/3} = a^{2/3}\) can be parametrized using equations involving cosine and sine. Let \(x = a\cos^3(t)\) and \(y = a\sin^3(t)\), where \(0 \leq t < 2\pi\). This parameterization covers all points on the astroid.

Step 2: Find Derivative for Tangent Slope

To find the tangent line, we need the slope, which is given by the derivative \(\frac{dy}{dx}\). Start with \(x = a\cos^3(t)\) and \(y = a\sin^3(t)\), differentiate each with respect to \(t\). We have \(\frac{dx}{dt} = -3a\cos^2(t)\sin(t)\) and \(\frac{dy}{dt} = 3a\sin^2(t)\cos(t)\). Thus, the slope \(\frac{dy}{dx}\) becomes \(\frac{\sin(t)}{\cos(t)} = \tan(t)\).

Step 3: Determine Equation of Tangent Line

The equation of the tangent line \(T\) at a point \((x_0, y_0)\) on the astroid is \(y - y_0 = m(x - x_0)\), where \(m = \tan(t)\) is the slope at that point. This simplifies to \(y - a\sin^3(t) = \tan(t)(x - a\cos^3(t))\).

Step 4: Find Intercepts on Axes

To find intercepts, set \(y = 0\) (for x-intercept) and \(x = 0\) (for y-intercept) one at a time. For the x-intercept, set \(y = 0\) and solve for \(x\): \(x = a\cos^3(t) - a\sin^3(t)\cot(t)\). For the y-intercept, set \(x = 0\) and solve for \(y\): \(y = a\sin^3(t) - a\cos^3(t)\tan(t)\).

Step 5: Calculate Distance Between Intercepts

The distance \(D\) between the x-intercept \(x = a(\cos^3(t) - \sin^3(t)\cot(t))\) and y-intercept \(y = a(\sin^3(t) - \cos^3(t)\tan(t))\) can be expressed as an absolute value formula. Notice that both terms simplify to \(a\sec(t)\csc(t)\). Hence, \(D = |a\sec(t) + a\csc(t)| = a |\sec(t) + \csc(t)|\).

Step 6: Analyze Dependence of Distance on T

The distance \(D = a |\sec(t) + \csc(t)|\) shows that \(D\) is directly proportional to the parameter \(t\) through its trigonometric relationship. As \(t\) varies, \(D\) changes accordingly depending on the value of \(\sec(t)\) and \(\csc(t)\).

Key Concepts

Parametric Equations

An astroid, a fascinating curve in mathematics, is best understood using parametric equations. These equations enable us to express the coordinates of points on the astroid in terms of a single variable, known as a parameter. For the astroid \[x^{2/3} + y^{2/3} = a^{2/3}\]the parametric equations are given as:

• \(x = a \cos^3(t)\)
• \(y = a \sin^3(t)\)

These equations cover every point on the astroid as the parameter \(t\) varies from 0 to \(2\pi\). This is akin to tracing out the complete shape of the astroid, which has a star-like appearance. Switching to parametric form not only simplifies calculations but also assists in visualizing the curve's geometry.

Tangent Line

A tangent line is a straight line that touches a curve at exactly one point. At this point of contact, the tangent line has the same slope as the curve itself. To determine the tangent line to an astroid at a given point, the key is to calculate the derivative of the parametric equations.From the given astroid parameterization, the derivatives with respect to \(t\) are:

• \(\frac{dx}{dt} = -3a\cos^2(t)\sin(t)\)
• \(\frac{dy}{dt} = 3a\sin^2(t)\cos(t)\)

The slope of the tangent line \(\frac{dy}{dx}\) is then found to be the ratio \(\tan(t)\). So, the equation of the tangent line at point \((x_0, y_0)\) is:\[y - a\sin^3(t) = \tan(t)(x - a\cos^3(t))\]This equation is fundamental in finding where the tangent intersects the axes, which helps analyze how it behaves relative to the astroid.

Intercepts

Intercepts are points where a line crosses the axes of a graph. For the tangent line to the astroid, finding these intercepts involves setting one of the coordinates to zero and solving for the other.For the x-intercept, where the line crosses the x-axis, set \(y = 0\). Solve the tangent line equation:\[x = a \cos^3(t) - a \sin^3(t) \cot(t)\]For the y-intercept, where the line crosses the y-axis, set \(x = 0\). Solve for:\[y = a \sin^3(t) - a \cos^3(t) \tan(t)\]These intercepts are crucial for understanding how far apart the line spans between the axes, giving insight into its length as it skims past the astroid curve.

Distance Formula

The distance formula is an essential tool in geometry, used to compute the distance between two points on a graph. For the tangent line intercepts on the astroid, the formula calculates the span between the x-intercept and y-intercept.The x-intercept is at \[x = a(\cos^3(t) - \sin^3(t)\cot(t))\] and the y-intercept at \[y = a(\sin^3(t) - \cos^3(t)\tan(t))\].The distance \(D\) between these intercepts is given by:\[D = |a\sec(t) + a\csc(t)| = a |\sec(t) + \csc(t)|\]This reveals that the distance is influenced by the trigonometric values of \(t\), specifically through the secant and cosecant functions, indicating that as \(t\) shifts, so does this linear measure.