The given parametric equations of the astroid are:

$x = a \cos^3 t$

$y = a \sin^3 t$

Step 1: Compute $\frac{dy}{dx}$ (Slope of the Tangent)

To find the slope of the tangent, we compute:

$\frac{dx}{dt} = \frac{d}{dt} (a \cos^3 t) = a \cdot 3 \cos^2 t (-\sin t) = -3a \cos^2 t \sin t$

$\frac{dy}{dt} = \frac{d}{dt} (a \sin^3 t) = a \cdot 3 \sin^2 t \cos t = 3a \sin^2 t \cos t$

Thus, the slope of the tangent line is:

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

$= \frac{3 \sin^2 t \cos t}{-3 \cos^2 t \sin t} = -\frac{\sin t}{\cos t} = -\tan t$

Step 2: Compute the Point at $t = \frac{\pi}{4}$

Substituting $t = \frac{\pi}{4}$:

$x = a \cos^3 \frac{\pi}{4} = a \left(\frac{\sqrt{2}}{2}\right)^3 = a \frac{2\sqrt{2}}{8} = \frac{a\sqrt{2}}{4}$

$y = a \sin^3 \frac{\pi}{4} = a \left(\frac{\sqrt{2}}{2}\right)^3 = \frac{a\sqrt{2}}{4}$

Thus, the point is:

$\left(\frac{a\sqrt{2}}{4}, \frac{a\sqrt{2}}{4} \right)$

Step 3: Find the Equation of the Tangent Line

At $t = \frac{\pi}{4}$:

$\text{slope of tangent} = -\tan \frac{\pi}{4} = -1$

Using the point-slope form of a line:

$y - y_1 = m (x - x_1)$

$y - \frac{a\sqrt{2}}{4} = -1 \left( x - \frac{a\sqrt{2}}{4} \right)$

$y - \frac{a\sqrt{2}}{4} = -x + \frac{a\sqrt{2}}{4}$

$y + x = \frac{a\sqrt{2}}{2}$

Thus, the equation of the tangent line is:

$x + y = \frac{a\sqrt{2}}{2}$

Step 4: Find the Equation of the Normal Line

The normal line is perpendicular to the tangent, so its slope is the negative reciprocal of $-1$, which is 1.

Using the point-slope form:

$y - \frac{a\sqrt{2}}{4} = 1 \left( x - \frac{a\sqrt{2}}{4} \right)$

$y - \frac{a\sqrt{2}}{4} = x - \frac{a\sqrt{2}}{4}$

$y - x = 0$

Thus, the equation of the normal line is:

$x - y = 0$

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Final Answer:

• Tangent line: $x + y = \frac{a\sqrt{2}}{2}$
• Normal line: $x - y = 0$

Would you like a detailed step-by-step explanation of any part?

Related Questions:

1. How do you derive the general equation of an astroid?
2. What is the geometric significance of the normal to an astroid?
3. How do you determine the envelope of tangent lines to an astroid?
4. Can an astroid be represented in implicit form?
5. What is the curvature of an astroid at a given point?

Tip:

The astroid is a special case of a hypocycloid, specifically a hypocycloid of four cusps, and can be described by the equation $x^{2/3} + y^{2/3} = a^{2/3}$.