The center of a geometric shape, such as a circle or an ellipse, is the point that is equidistant from all points on the shape's perimeter. It is the point around which the shape is symmetrical and balanced.
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The center of a circle is the point that is equidistant from all points on the circle's circumference.
For an ellipse, the center is the point where the major and minor axes intersect, and it is the midpoint of the ellipse.
The center of a hyperbola is the point where the transverse and conjugate axes intersect, and it is the midpoint of the hyperbola.
When the axes of a conic section are rotated, the center of the shape remains the same, but the orientation of the axes changes.
Knowing the center of a conic section is crucial for understanding its properties, such as the shape, size, and orientation.
Review Questions
Explain the role of the center in the equation of a circle.
The center of a circle is the point $(h, k)$ that is used in the standard equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, where $r$ is the radius of the circle. The center represents the point around which the circle is symmetrical, and it is the point that is equidistant from all points on the circle's circumference.
Describe how the center of an ellipse is related to the major and minor axes.
The center of an ellipse is the point $(h, k)$ where the major and minor axes intersect. The major axis, $2a$, and the minor axis, $2b$, are both centered at the point $(h, k)$. The equation of an ellipse in standard form is $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, where $(h, k)$ represents the center of the ellipse.
Explain how the center of a hyperbola is affected by the rotation of the axes.
When the axes of a hyperbola are rotated, the center of the hyperbola remains the same, but the orientation of the axes changes. The center of the hyperbola is the point $(h, k)$ where the transverse and conjugate axes intersect. The equation of a hyperbola in standard form is $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$, where $(h, k)$ represents the center of the hyperbola. Rotating the axes changes the values of $a$ and $b$, but the center $(h, k)$ remains constant.