Unveiling the Secrets of (x+2)^2/9 + (y3)^2/16 = 1
This equation represents a conic section specifically an ellipse. Let's explore its features and how to interpret its form:
Understanding the Equation
The standard form of an ellipse centered at (h,k) is:
(xh)^2/a^2 + (yk)^2/b^2 = 1
Comparing this to our equation:
(x+2)^2/9 + (y3)^2/16 = 1
We can identify the following:
 Center: (h,k) = (2,3)
 Major axis: The larger denominator (16) corresponds to the yterm, indicating the major axis is vertical with length 2b = 8. Therefore, b = 4.
 Minor axis: The smaller denominator (9) corresponds to the xterm, indicating the minor axis is horizontal with length 2a = 6. Therefore, a = 3.
Key Features of the Ellipse:

Vertices: The vertices lie on the major axis, a distance of b units above and below the center. Therefore, the vertices are (2, 3 + 4) = (2, 7) and (2, 3  4) = (2, 1).

Covertices: The covertices lie on the minor axis, a distance of a units to the left and right of the center. Therefore, the covertices are (2 + 3, 3) = (1, 3) and (2  3, 3) = (5, 3).

Foci: The foci lie on the major axis, a distance of c units above and below the center, where c is calculated using the relationship: c^2 = b^2  a^2. In our case, c^2 = 16  9 = 7, so c = โ7. Therefore, the foci are (2, 3 + โ7) and (2, 3  โ7).
Visualizing the Ellipse
To visualize the ellipse, plot the center, vertices, covertices, and foci. Then, sketch a smooth curve through these points, maintaining the proportions determined by the major and minor axes.
Applications of Ellipses
Ellipses have numerous applications in various fields:
 Astronomy: Planetary orbits around the sun are elliptical.
 Engineering: Elliptical gears are used in machinery to achieve smooth, continuous motion.
 Architecture: Elliptical arches are used in construction for their strength and aesthetic appeal.
 Optics: Ellipses are used in designing lenses and reflectors for focusing light.
Understanding the equation and its features allows us to analyze and utilize the ellipse in various realworld applications.