show that the circles x^2 + y^2 - 10x - 8y + 18 = 0 and x^2 + y^2 - 8x - 4y + 14 = 0 do not intersect.
Please
Finally, even after you changed your name 3 times
first subtract them:
-2x -4y + 4=0
x + 2y + 2 =0
x = -2-2y
sub into the first:
(-2-2y)^2 + y^2 - 10(-2-2y) - 8y +18=0
4 + 8y + 4y^2 + y^2 + 20 + 20y - 8y + 18 = 0
5y^2 + 20y + 38=0
y = (-20 + ?-360)/10 , which is not a real number. So they can't intersect
see Wolfram, they don't meet
http://www.wolframalpha.com/input/?i=plot+x%5E2+%2B+y%5E2+-+10x+-+8y+%2B+18+%3D+0+and+x%5E2+%2B+y%5E2+-+8x+-+4y+%2B+14+%3D+0
To show that the circles do not intersect, we can compare their equations and analyze their centers and radii.
The general equation of a circle is given by:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle and r represents the radius.
Let's compare the equations of the two circles:
Circle 1: x^2 + y^2 - 10x - 8y + 18 = 0
Circle 2: x^2 + y^2 - 8x - 4y + 14 = 0
To determine the centers, we need to rewrite the equations in the standard form:
Circle 1: (x^2 - 10x) + (y^2 - 8y) = -18
Circle 2: (x^2 - 8x) + (y^2 - 4y) = -14
Completing the square for x and y terms, we get:
Circle 1: (x^2 - 10x + 25) + (y^2 - 8y + 16) = -18 + 25 + 16
Circle 2: (x^2 - 8x + 16) + (y^2 - 4y + 4) = -14 + 16 + 4
Simplifying, we have:
Circle 1: (x - 5)^2 + (y - 4)^2 = 23
Circle 2: (x - 4)^2 + (y - 2)^2 = 6
Comparing the centers, we see that the center of Circle 1 is at (5, 4), and the center of Circle 2 is at (4, 2).
Now, let's compare the radii:
The radius of Circle 1 is sqrt(23), while the radius of Circle 2 is sqrt(6).
Since the two circles have different centers and different radii, they do not intersect.
To show that two circles do not intersect, we need to check for any overlap of their respective areas. In other words, we need to find if there is any common solution to the equations of the circles.
Let's first write the equations of the given circles in a standard form:
Circle 1: x^2 + y^2 - 10x - 8y + 18 = 0
Circle 2: x^2 + y^2 - 8x - 4y + 14 = 0
Now, we can compare the equations of the circles. By subtracting the equations, we can determine the equation of the line that separates the two circles.
(x^2 + y^2 - 10x - 8y + 18) - (x^2 + y^2 - 8x - 4y + 14) = 0
Simplifying, we get:
-10x - 8y + 18 + 8x + 4y - 14 = 0
-2x - 4y + 4 = 0
-x - 2y + 2 = 0
2y = -x + 2
y = (-1/2)x + 1
We now have the equation of the line that separates the two circles. Now, we need to analyze the positions of the circles with respect to this line.
To determine the relative positions of the circles and the line, we can check the distance between the center of each circle and the line. The distance between a point (x1, y1) and a line Ax + By + C = 0 is given by the formula:
d = |Ax1 + By1 + C| / sqrt(A^2 + B^2)
Let's find the distances from the centers of the circles to the line y = (-1/2)x + 1:
1. Circle 1:
Center: (5, 4)
Distance: |(-1/2)(5) + (1)(4) - 2| / sqrt((-1/2)^2 + 1^2)
= |(-5/2) + 4 - 2| / sqrt(1/4 + 1)
= |(-5/2) + 2| / sqrt(5/4 + 4/4)
= |(-5/2) + 4/2| / sqrt(9/4)
= |-1/2| / (3/2)
= 1 / (3/2)
= 2/3
2. Circle 2:
Center: (4, 2)
Distance: |(-1/2)(4) + (1)(2) - 2| / sqrt((-1/2)^2 + 1^2)
= |(-4/2) + 2 - 2| / sqrt(1/4 + 1)
= |(-2) + 2 - 2| / sqrt(1/4 + 4/4)
= |(-2)| / sqrt(5/4)
= 2 / (sqrt(5)/2)
= 4 / sqrt(5)
= 4 sqrt(5) / 5
Now, let's compare the distances:
Distance between the centers of the circles:
sqrt((5-4)^2 + (4-2)^2) = sqrt(1 + 4) = sqrt(5) > 0
Since the distance between the centers of the circles is greater than zero and both distances are positive, the circles are separated. Therefore, the circles do not intersect.
To summarize:
- We found the equation of the line that separates the circles by subtracting the equations.
- We calculated the distances from the centers of the circles to the separating line.
- By comparing the distances and observing that they are both positive and non-zero, we concluded that the circles do not intersect.