Conic Sections Question 5

One of the following equations represents the ellipse shown below in the standard (x,y) coordinate plane. Which one?

\displaystyle \frac{(x-2)^2}{3}+ \frac{(y+3)^2}{5} = 1

\displaystyle \frac{(x+2)^2}{3}+ \frac{(y-3)^2}{5} = 1

\displaystyle \frac{(x+2)^2}{5}+ \frac{(y-3)^2}{3} = 1

\displaystyle \frac{(x-2)^2}{9}+ \frac{(y+3)^2}{25} = 1

\displaystyle \frac{(x+2)^2}{9}+ \frac{(y-3)^2}{25} = 1

Approach

The equation of an ellipse:

\frac{(x-h)^2}{a^2}+ \frac{(y-k)^2}{b^2} = 1

$(h,k)$ represents the center. $a$ represents the length from the center to one of the points in the horizontal direction. $b$ represents the distance from the center to a point directly above or below. Based on the graph,

(h,k) = (2,-3)

a= 3

b=5

\frac{(x-2)^2}{(3)^2}+ \frac{(y-(-3))^2}{(5)^2} = 1

\boxed{ \frac{(x-2)^2}{9}+ \frac{(y+3)^2}{25} = 1 }