91Ó°ÊÓ

Parabolas are unique conic sections that can be identified by their characteristic shape—a U or an inverted U. They can open upwards, downwards, to the left or to the right. The standard forms of a parabola are \( y^2 = 4ax \) and \( x^2 = 4ay \). When the parabola opens to the side, the equation might be written as \( x = ay^2 + c \), where adjusting the coefficients shifts the orientation and position of the parabola.

In the given exercise, we can see an example in Equation c: \( y - x^2 = 5 \), which can be rearranged to \( y = x^2 + 5 \), showing a parabola opening upwards. Also, Equation e: \( x - 2y^2 = 3 \) can be rewritten as \( x = 2y^2 + 3 \), indicating a parabola opening to the right.

An ellipse can be visualized as an elongated circle or an oval. It is a conic section that is formed by slicing through a cone at an angle. The standard form equation of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the distances from the center to the ellipse along the x and y axes, respectively.

In the exercise, Equation a: \( \frac{x^2}{10} + \frac{y^2}{12} = 1 \) matches the form of an ellipse. Similarly, Equation h: \( \frac{(x-1)^2}{10} + \frac{(y-4)^2}{8} = 1 \) is a translated ellipse centered at (1, 4).

Hyperbolas consist of two separate, mirror-image curves that open away from each other. The standard form of a hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), depending on the orientation. Hyperbolas have asymptotes that the curves approach but never touch.

In the exercise, Equation d: \( \frac{x^2}{9} - \frac{y^2}{10} = 1 \) is a hyperbola. Equation f: \( \frac{y^2}{20} - \frac{x^2}{35} = 1 \) is also in the standard hyperbola form but oriented differently.

A circle is a special type of ellipse where the distances from the center to any point on the circumference are constant. The standard equation of a circle is \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle.

In the exercise, Equation b: \( (x + 1)^2 + (y - 3)^2 = 30 \) represents a circle with a radius of \( \sqrt{30} \), centered at (-1, 3). Equation g: \( 3x^2 + 3y^2 = 75 \) can be simplified by dividing by 3 to get \( x^2 + y^2 = 25 \), indicating a circle with a radius of 5.