(x+2)^2/9+(y-3)^2/16=1

4 min read Jun 16, 2024

Unveiling the Secrets of (x+2)^2/9 + (y-3)^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:

(x-h)^2/a^2 + (y-k)^2/b^2 = 1

Comparing this to our equation:

(x+2)^2/9 + (y-3)^2/16 = 1

We can identify the following:

Center: (h,k) = (-2,3)
Major axis: The larger denominator (16) corresponds to the y-term, indicating the major axis is vertical with length 2b = 8. Therefore, b = 4.
Minor axis: The smaller denominator (9) corresponds to the x-term, 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).
Co-vertices: The co-vertices lie on the minor axis, a distance of a units to the left and right of the center. Therefore, the co-vertices 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 = √7. Therefore, the foci are (-2, 3 + √7) and (-2, 3 - √7).

Visualizing the Ellipse

To visualize the ellipse, plot the center, vertices, co-vertices, 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 real-world applications.