show that the circles x^2 + y^2 - 10x - 8y + 18 = 0 and x^2 + y^2 - 8y - 4y +14x = 0 do not intersect
check your typing, the second circle contains two y terms and no constant.
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To show that the circles given by the equations:
1. x^2 + y^2 - 10x - 8y + 18 = 0
2. x^2 + y^2 - 8y - 4x + 14x = 0
do not intersect, we need to find their centers and radii.
Step 1: Convert the equations to standard form.
1. x^2 + y^2 - 10x - 8y + 18 = 0
Rewrite as:
(x^2 - 10x) + (y^2 - 8y) + 18 = 0
Complete the square by adding and subtracting half of the coefficients of x and y:
(x^2 - 10x + 25) + (y^2 - 8y + 16) + 18 - 25 - 16 = 0
Simplify:
(x - 5)^2 + (y - 4)^2 - 27 = 0
2. x^2 + y^2 - 8y - 4x + 14x = 0
Rewrite as:
(x^2 - 4x) + (y^2 - 8y) + 0 = 0
Complete the square by adding and subtracting half of the coefficients of x and y:
(x^2 - 4x + 4) + (y^2 - 8y +16) - 4 - 16 = 0
Simplify:
(x - 2)^2 + (y - 4)^2 - 4 - 16 = 0
(x - 2)^2 + (y - 4)^2 - 20 = 0
Step 2: Compare the equations to the standard form of a circle: (x - h)^2 + (y - k)^2 = r^2
1. The center of the first circle is (5, 4) and the radius is sqrt(27).
2. The center of the second circle is (2, 4) and the radius is sqrt(20).
Step 3: Determine if the circles intersect.
The distance between the centers of the two circles can be calculated as:
d = sqrt((2-5)^2 + (4-4)^2)
= sqrt(9 + 0)
= sqrt(9)
= 3
Since the sum of the radii (sqrt(27) + sqrt(20)) is greater than the distance between the centers (3), the two circles do not intersect.
To show that two circles do not intersect, we can check if the distance between their centers is greater than the sum of their radii.
The general form of a circle equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.
Let's find the equation of the circles in the given question:
1) Circle 1: x^2 + y^2 - 10x - 8y + 18 = 0
Rearrange the equation: (x^2 - 10x) + (y^2 - 8y) = -18
Complete the square for x: (x^2 - 10x + 25) + (y^2 - 8y) = -18 + 25
Complete the square for y: (x^2 - 10x + 25) + (y^2 - 8y + 16) = -18 + 25 + 16
Simplify: (x - 5)^2 + (y - 4)^2 = 23
2) Circle 2: x^2 + y^2 - 8y - 4y + 14x = 0
Rearrange the equation: (x^2 + 14x) + (y^2 - 8y - 4y) = 0
Complete the square for x: (x^2 + 14x + 49) + (y^2 - 8y - 4y) = 49
Complete the square for y: (x^2 + 14x + 49) + (y^2 - 8y - 4y + 16) = 49 + 16
Simplify: (x + 7)^2 + (y - 6)^2 = 65
From the equations above, we can see that the centers of the circles are (5, 4) for Circle 1 and (-7, 6) for Circle 2.
Now, let's calculate the distance between the two centers:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt(((-7) - 5)^2 + (6 - 4)^2)
= sqrt((-12)^2 + 2^2)
= sqrt(144 + 4)
= sqrt(148)
= 2 √37
Next, we find the radii of the circles:
Radius of Circle 1 = sqrt(23)
Radius of Circle 2 = sqrt(65)
Finally, we compare the distance between the centers and the sum of the radii:
Distance > Radius of Circle 1 + Radius of Circle 2
2 √37 > sqrt(23) + sqrt(65)
Since the distance between the centers is greater than the sum of the radii, the two circles do not intersect.