In the standard (x, y) coordinate plane, two circles are defined by the equations x^2 + y^… — ACT

In the standard $(x, y)$ coordinate plane, two circles are defined by the equations $x^2 + y^2 - 4x + 6y = 3$ and $x^2 + y^2 + 8x - 2y = 8$ . Is the statement that the line passing through the centers of these two circles has a slope of $-\frac{2}{3}$ true or false?

Answer: Answer

Answer

true

The correct answer is true because completing the square for the given circle equations determines the centers to be

(2, -3)

and

(-4, 1)

. Calculating the slope between these two coordinates using the slope formula results in

-\frac{2}{3}

, which matches the statement.

Step-by-Step Solution

Complete the square for the first circle's equation to find its center.

The equation

x^2 + y^2 - 4x + 6y = 3

is rewritten as

(x - 2)^2 + (y + 3)^2 = 16

, identifying the center of the first circle as

(2, -3)

Converting the circle's equation to the standard form

(x - h)^2 + (y - k)^2 = r^2

reveals the center coordinates

(h, k)

Complete the square for the second circle's equation to find its center.

The equation

x^2 + y^2 + 8x - 2y = 8

is rewritten as

(x + 4)^2 + (y - 1)^2 = 25

, identifying the center of the second circle as

(-4, 1)

Converting the second circle's equation to standard form provides the coordinates of the second point needed to determine the slope.

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

to the two center points.

Using the coordinates

(2, -3)

and

(-4, 1)

, we calculate

m = \frac{1 - (-3)}{-4 - 2} = \frac{4}{-6} = -\frac{2}{3}

This calculation determines the slope of the line passing through both centers to evaluate the statement.

Key Concept

Slope of a Line

Question

Answer

Step-by-Step Solution

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