The task involves describing and sketching the sets $A$, $B$, $C$, and $D$ for given parameters. Let's go through each set step-by-step to understand and sketch them mathematically:

(a) Set $A$

$A := \left\{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = r^2 \right\}$

This equation represents a circle centered at the origin with radius $r$. Here:

• The parameter $r$ defines the radius, with $r > 0$.
• Points $(x, y)$ satisfying $x^2 + y^2 = r^2$ lie on the circumference of this circle.

Sketch:

• A circle centered at the origin with radius $r$.

(b) Set $B$

$B := \left\{ (x, y) \in \mathbb{R}^2 \mid \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \right\}$

This equation represents an ellipse centered at the origin.

• The semi-major axis is $a$ along the $x$-axis.
• The semi-minor axis is $b$ along the $y$-axis.

Sketch:

• An ellipse centered at the origin, with its width determined by $a$ along the $x$-axis and height by $b$ along the $y$-axis.

(c) Set $C$

$C := \left\{ (x, y) \in \mathbb{R}^2 \mid \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \right\}$

This equation represents a hyperbola centered at the origin.

• If $a$ and $b$ are positive, the hyperbola opens along the $x$-axis.
• The vertices of the hyperbola are located at $(\pm a, 0)$.

Sketch:

• A hyperbola centered at the origin, opening along the $x$-axis, with vertices at $(\pm a, 0)$.

(d) Set $D$

$D := \left\{ (x, y) \in \mathbb{R}^2 \mid y^2 = ax \right\}$

This equation describes a parabola.

• The parabola opens to the right if $a > 0$, and to the left if $a < 0$.
• The vertex of this parabola is at the origin.

Sketch:

• A parabola centered at the origin, opening to the right for $a > 0$ or to the left for $a < 0$.

Would you like a specific visualization of these sketches or further details on each shape?

Follow-up Questions

1. How does the value of $a$ and $b$ affect the shape of the ellipse and hyperbola?
2. What happens to the parabola in set $D$ if $a$ is negative?
3. Could the radius $r$ in set $A$ be related to the parameters $a$ and $b$ in the other sets?
4. How can we determine the asymptotes of the hyperbola in set $C$?
5. What are real-world applications of each conic section represented in sets $A$ to $D$?

Tip

For conic sections, each parameter (like $a$, $b$, and $r$) uniquely influences the size, orientation, and shape of the figure.