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Zorluk: ZorSlope of a Line

In the standard (x,y)(x, y) coordinate plane, two circles are defined by the equations x2+y2−4x+6y=3x^2 + y^2 - 4x + 6y = 3 and x2+y2+8x−2y=8x^2 + y^2 + 8x - 2y = 8. Is the statement that the line passing through the centers of these two circles has a slope of −23-\frac{2}{3} true or false?

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true
The correct answer is true because completing the square for the given circle equations determines the centers to be (2,−3)(2, -3) and (−4,1)(-4, 1). Calculating the slope between these two coordinates using the slope formula results in −23-\frac{2}{3}, which matches the statement.

Adım Adım Çözüm

1
Complete the square for the first circle's equation to find its center.
The equation x2+y2−4x+6y=3x^2 + y^2 - 4x + 6y = 3 is rewritten as (x−2)2+(y+3)2=16(x - 2)^2 + (y + 3)^2 = 16, identifying the center of the first circle as (2,−3)(2, -3).
Converting the circle's equation to the standard form (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 reveals the center coordinates (h,k)(h, k).
2
Complete the square for the second circle's equation to find its center.
The equation x2+y2+8x−2y=8x^2 + y^2 + 8x - 2y = 8 is rewritten as (x+4)2+(y−1)2=25(x + 4)^2 + (y - 1)^2 = 25, identifying the center of the second circle as (−4,1)(-4, 1).
Converting the second circle's equation to standard form provides the coordinates of the second point needed to determine the slope.
3
Apply the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} to the two center points.
Using the coordinates (2,−3)(2, -3) and (−4,1)(-4, 1), we calculate m=1−(−3)−4−2=4−6=−23m = \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.

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Slope of a Line
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Slope of a Line

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