show that the circles x^2 + y^2 - 10x - 8y + 18 = 0 and x^2 + y^2 - 8y - 4y + 14x = 0 do not intersect
x^2 + y^2 - 10x - 8y + 18 = 0
x^2-10x + y^2-8y = -18
x^2-10x+25 + y^2-8y+16 = -18+25+16
(x-5)^2 + (y-4)^2 = 23
The second equation has a typo. Fix it and complete the squares to find its center.
Then compare the distance between centers to the sum of the radii.
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To prove that the two circles do not intersect, we can compare their equations and analyze their key characteristics. Here are the steps to demonstrate that the circles do not intersect:
Step 1: Write the equations of the circles in their standard form.
The equation x^2 + y^2 - 10x - 8y + 18 = 0 can be written as:
(x - 5)^2 + (y - 4)^2 = 5^2
The equation x^2 + y^2 - 8y - 4y + 14x = 0 can be written as:
(x + 7)^2 + (y - 4)^2 = 3^2
Step 2: Compare the centers of the circles.
The center of the first circle is (5, 4), and the center of the second circle is (-7, 4).
Since the y-coordinate of both centers is the same (4), we can establish that both circles are aligned horizontally.
Step 3: Calculate the distance between the centers.
Using the distance formula, we can find the distance between the centers of the circles:
Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2]
Distance = sqrt[(-7 - 5)^2 + (4 - 4)^2]
Distance = sqrt[(-12)^2]
Distance = 12
Step 4: Compare the radii of the circles.
The radius of the first circle is 5, and the radius of the second circle is 3.
Step 5: Analyze the relationship between the centers and the radii.
Since the distance between the centers (12) is greater than the sum of the radii (5 + 3 = 8), we conclude that the circles do not intersect.
Therefore, by comparing the centers, radii, and distances between them, we can determine that the circles x^2 + y^2 - 10x - 8y + 18 = 0 and x^2 + y^2 - 8y - 4y + 14x = 0 do not intersect.