An ellipse in the standard (x, y) coordinate plane is defined by the equation \frac{(x-2)^… — ACT

An ellipse in the standard $(x, y)$ coordinate plane is defined by the equation

\frac{(x-2)^2}{9} + \frac{(y+5)^2}{16} = 1

What are the coordinates of the center of this ellipse?

A
$(-2, 5)$
B
$(2, 5)$
$(2, -5)$ Answer
D
$(-2, -5)$
E
$(-5, 2)$

The center of the ellipse is

(2, -5)

The standard form of an ellipse equation is

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

, where the center of the ellipse is at the coordinate point

(h, k)

. Comparing the given equation

\frac{(x-2)^2}{9} + \frac{(y+5)^2}{16} = 1

to the standard form reveals that

h = 2

and

k = -5

. Therefore, the center coordinates are

(2, -5)

Recall the standard form equation of an ellipse with horizontal/vertical axes.

The standard form is

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

, where

(h, k)

represents the coordinates of the center.

This establishes the framework to extract the center coordinate values.

Compare the terms in the given equation to the standard form.

Matching

(x-h)^2

with

(x-2)^2

gives

h = 2

. Matching

(y-k)^2

with

(y+5)^2 = (y-(-5))^2

gives

k = -5

Comparing terms identifies the offsets

h

and

k

that determine the center.

Write the center coordinate pair

(h, k)

The center is

(2, -5)

Combining the values of

h

and

k

yields the final coordinates.

Identifying the center of an ellipse from its standard form equation

Estimated Time:45s

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