Question

What is the equation of the ellipse with co-vertices (0, 2), (0, -2), and vertices (3, 0), (-3, 0)?

a) ( x²/9 + y²/4 = 1 )
b) ( x²/3 + y²/4 = 1 )
c) ( x²/9 + y²/16 = 1 )
d) ( x²/3 + y²/16 = 1 )

Answer 1

The equation of the ellipse with the given co-vertices and vertices is \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), which corresponds to option (a).

Step-by-step explanation:

The student is asking for the equation of an ellipse given its co-vertices and vertices. An ellipse is generally represented by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where 2a is the length of the major axis and 2b is the length of the minor axis. In this case, we have vertices at (3,0) and (-3,0), which implies that the length of the major axis is 2a = 6; hence, a = 3. The co-vertices at (0,2) and (0,-2) imply that the length of the minor axis is 2b = 4; thus, b = 2. Substituting a = 3 and b = 2 into the standard equation gives us \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), which is option (a).