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### 1) Difference Between the Equations for Parabolas, Ellipses, Circles, and Hyperbolas (10 points)
#### Parabola:
- Equation Form: The standard form of a parabola can generally be written in two ways based on the orientation:
- Vertical orientation: [tex]\( y = ax^2 + bx + c \)[/tex]
- Horizontal orientation: [tex]\( x = ay^2 + by + c \)[/tex]
In a parabola, the squared term represents the variable that's being squared, and the other term is linear. The coefficient values determine the shape and orientation of the parabola.
#### Ellipse:
- Equation Form: The standard form of an ellipse is:
[tex]\[ \left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) = 1 \][/tex]
Here [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the semi-major and semi-minor axes of the ellipse, respectively. The ellipse is a conic section that appears as a squished circle.
#### Circle:
- Equation Form: The standard form of a circle is:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
In this case, [tex]\( r \)[/tex] represents the radius of the circle. A circle can be considered a special case of an ellipse where [tex]\( a = b \)[/tex] or where the semi-major and semi-minor axes are equal.
#### Hyperbola:
- Equation Form: The standard form of a hyperbola can vary depending on the orientation of the transverse axis. The two forms are:
- Hyperbola with a horizontal transverse axis:
[tex]\[ \left(\frac{x^2}{a^2}\right) - \left(\frac{y^2}{b^2}\right) = 1 \][/tex]
- Hyperbola with a vertical transverse axis:
[tex]\[ \left(\frac{y^2}{a^2}\right) - \left(\frac{x^2}{b^2}\right) = 1 \][/tex]
In a hyperbola, the difference between the squares of the distances to two fixed points (foci) is constant. The hyperbola consists of two disconnected curves called branches.
### 2) Standard Equations of Hyperbola with a Horizontal and Vertical Transverse Axis (10 points)
#### Hyperbola with a Horizontal Transverse Axis:
The standard equation for a hyperbola with a horizontal transverse axis is:
[tex]\[ \left(\frac{x^2}{a^2}\right) - \left(\frac{y^2}{b^2}\right) = 1 \][/tex]
In this form, the transverse axis is along the x-axis.
#### Hyperbola with a Vertical Transverse Axis:
The standard equation for a hyperbola with a vertical transverse axis is:
[tex]\[ \left(\frac{y^2}{a^2}\right) - \left(\frac{x^2}{b^2}\right) = 1 \][/tex]
In this form, the transverse axis is along the y-axis.
By comparing these forms, we can see that the primary difference depends on which squared term is positive. In the horizontal case, [tex]\( x^2 \)[/tex] is positive, whereas, in the vertical case, [tex]\( y^2 \)[/tex] is positive.
These equations provide the basic framework for understanding the geometry of hyperbolas in relation to their axes orientations.
### 1) Difference Between the Equations for Parabolas, Ellipses, Circles, and Hyperbolas (10 points)
#### Parabola:
- Equation Form: The standard form of a parabola can generally be written in two ways based on the orientation:
- Vertical orientation: [tex]\( y = ax^2 + bx + c \)[/tex]
- Horizontal orientation: [tex]\( x = ay^2 + by + c \)[/tex]
In a parabola, the squared term represents the variable that's being squared, and the other term is linear. The coefficient values determine the shape and orientation of the parabola.
#### Ellipse:
- Equation Form: The standard form of an ellipse is:
[tex]\[ \left(\frac{x^2}{a^2}\right) + \left(\frac{y^2}{b^2}\right) = 1 \][/tex]
Here [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the semi-major and semi-minor axes of the ellipse, respectively. The ellipse is a conic section that appears as a squished circle.
#### Circle:
- Equation Form: The standard form of a circle is:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
In this case, [tex]\( r \)[/tex] represents the radius of the circle. A circle can be considered a special case of an ellipse where [tex]\( a = b \)[/tex] or where the semi-major and semi-minor axes are equal.
#### Hyperbola:
- Equation Form: The standard form of a hyperbola can vary depending on the orientation of the transverse axis. The two forms are:
- Hyperbola with a horizontal transverse axis:
[tex]\[ \left(\frac{x^2}{a^2}\right) - \left(\frac{y^2}{b^2}\right) = 1 \][/tex]
- Hyperbola with a vertical transverse axis:
[tex]\[ \left(\frac{y^2}{a^2}\right) - \left(\frac{x^2}{b^2}\right) = 1 \][/tex]
In a hyperbola, the difference between the squares of the distances to two fixed points (foci) is constant. The hyperbola consists of two disconnected curves called branches.
### 2) Standard Equations of Hyperbola with a Horizontal and Vertical Transverse Axis (10 points)
#### Hyperbola with a Horizontal Transverse Axis:
The standard equation for a hyperbola with a horizontal transverse axis is:
[tex]\[ \left(\frac{x^2}{a^2}\right) - \left(\frac{y^2}{b^2}\right) = 1 \][/tex]
In this form, the transverse axis is along the x-axis.
#### Hyperbola with a Vertical Transverse Axis:
The standard equation for a hyperbola with a vertical transverse axis is:
[tex]\[ \left(\frac{y^2}{a^2}\right) - \left(\frac{x^2}{b^2}\right) = 1 \][/tex]
In this form, the transverse axis is along the y-axis.
By comparing these forms, we can see that the primary difference depends on which squared term is positive. In the horizontal case, [tex]\( x^2 \)[/tex] is positive, whereas, in the vertical case, [tex]\( y^2 \)[/tex] is positive.
These equations provide the basic framework for understanding the geometry of hyperbolas in relation to their axes orientations.
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