An ellipse in the standard (x, y) coordinate plane is defined by the equation 25x^2 + 9y^2… — ACT

An ellipse in the standard $(x, y)$ coordinate plane is defined by the equation $25x^2 + 9y^2 - 100x + 54y - 44 = 0$ . Which of the following points is a focus of this ellipse?

A
$(-2, 7)$
B
$(2, -1)$
$(2, 1)$ Answer
D
$(6, -3)$
E
$(2, 19)$

Answer

(2, 1)

The correct answer is the point

(2, 1)

because rewriting the general equation of the ellipse in standard form gives

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

. The center is

(2, -3)

and the major axis is vertical with a focal distance of

c = \sqrt{25 - 9} = 4

. Adding this distance to the

y

-coordinate of the center yields the focus

(2, -3 + 4) = (2, 1)

Step-by-Step Solution

Group the

x

and

y

terms and factor out the coefficients.

25(x^2 - 4x) + 9(y^2 + 6y) = 44

This prepares the algebraic equation for completing the square.

Complete the square for both the

x

and

y

groups.

25(x-2)^2 + 9(y+3)^2 = 225

Adding

25 \times 4 = 100

and

9 \times 9 = 81

to both sides maintains equality while converting the quadratic expressions into perfect square trinomials.

Divide both sides by

225

to write the equation in standard form.

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

The standard form of an ellipse equation is

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

(for a vertical major axis), which directly reveals the center

(h, k)

and the axis parameters.

Identify the center, axis lengths, and calculate the focal distance

c

Center is

(2, -3)

a^2 = 25

b^2 = 9

. Thus,

c = \sqrt{25 - 9} = 4

The focal distance

c

for an ellipse is determined by the relation

c = \sqrt{a^2 - b^2}

Determine the coordinates of the foci.

Foci are

(2, -3 \pm 4)

, which simplifies to

(2, 1)

and

(2, -7)

Since

a^2 = 25

is under the

y

-term, the ellipse is vertically oriented, meaning the foci lie on the vertical line passing through the center.

Key Concept

Rewriting the general equation of an ellipse to standard form and finding its foci.

Estimated Time:2m 30s

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