What is the shortest distance between the circles defined by x^2-10x +y^2-4y-7=0 and x^2+14x +y^2+6y+49=0?
check what I just answered.
http://www.jiskha.com/display.cgi?id=1460217578
To find the shortest distance between the circles, we need to find their common tangent line and then find the distance between that tangent line and the center of each circle.
Step 1: Convert the equations of the circles to standard form:
For the first circle, we complete the square by adding and subtracting the necessary terms:
(x^2 - 10x + 25) + (y^2 - 4y + 4) - 7 - 25 + 4 = 0
(x - 5)^2 + (y - 2)^2 - 28 = 0
(x - 5)^2 + (y - 2)^2 = 28
For the second circle, we complete the square by adding and subtracting the necessary terms:
(x^2 + 14x + 49) + (y^2 + 6y + 9) - 49 - 49 + 9 = 0
(x + 7)^2 + (y + 3)^2 - 89 = 0
(x + 7)^2 + (y + 3)^2 = 89
Step 2: Find the centers and radii of the circles:
For the first circle, the center is (5, 2) and the radius is √28.
For the second circle, the center is (-7, -3) and the radius is √89.
Step 3: Find the common tangent line between the circles:
The common tangent line is perpendicular to the line connecting the centers of the circles.
Using the midpoint formula, we can find the line connecting the centers:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint = ((5 + (-7))/2, (2 + (-3))/2)
Midpoint = (-1, -0.5)
Using the slope formula, we can find the slope of the line connecting the centers:
Slope = (y2 - y1)/(x2 - x1)
Slope = (-3 - 2)/(-7 - 5)
Slope = -5/(-12)
Slope = 5/12
The slope of the common tangent line will be the negative reciprocal of the slope of the line connecting the centers. So, the slope of the common tangent line is -12/5.
Using the point-slope form, we can find the equation of the common tangent line:
y - (-0.5) = -12/5(x - (-1))
y + 0.5 = -12/5(x + 1)
5y + 2.5 = -12x - 12
5y = -12x - 14.5
12x + 5y + 14.5 = 0
Step 4: Find the distance between the tangent line and the centers of the circles:
The distance between a line and a point can be calculated using the formula:
Distance = |Ax + By + C| / √(A^2 + B^2)
For the first circle:
A = 12, B = 5, C = 14.5, x1 = 5, y1 = 2
Distance = |12(5) + 5(2) + 14.5| / √(12^2 + 5^2)
Distance = |60 + 10 + 14.5| / √(144 + 25)
Distance = 84.5 / √169
Distance = 84.5 / 13
Distance ≈ 6.50
For the second circle:
A = 12, B = 5, C = 14.5, x1 = -7, y1 = -3
Distance = |12(-7) + 5(-3) + 14.5| / √(12^2 + 5^2)
Distance = |-84 - 15 + 14.5| / √(144 + 25)
Distance = |-84 - 15 + 14.5| / √169
Distance = |-84 - 0.5| / 13
Distance ≈ 6.50
Therefore, the shortest distance between the circles is approximately 6.50 units.
To find the shortest distance between the circles defined by the equations x^2-10x +y^2-4y-7=0 and x^2+14x +y^2+6y+49=0, we need to find the distance between their centers minus the sum of their radii.
Step 1: Rewrite the equations in standard form
Completing the square will help us write the equations in standard form ((x - h)^2 + (y - k)^2 = r^2), where (h, k) is the center of the circle and r is the radius.
x^2 - 10x + y^2 - 4y - 7 = 0
(x^2 - 10x + 25) + (y^2 - 4y + 4) - 7 - 25 + 4 = 25 + 4 - 7
(x - 5)^2 + (y - 2)^2 = 22
x^2 + 14x + y^2 + 6y +49 = 0
(x^2 + 14x + 49) + (y^2 + 6y + 9) - 49 - 9 + 49 = 49 + 9 - 49
(x + 7)^2 + (y + 3)^2 = 9
Now we can identify the centers and radii of the circles.
For the first circle:
Center: (5, 2)
Radius: √22
For the second circle:
Center: (-7, -3)
Radius: 3
Step 2: Find the distance between the centers of the circles
Using the distance formula, we can calculate the distance between the centers.
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Distance = √[(-7 - 5)^2 + (-3 - 2)^2]
Distance = √[(-12)^2 + (-5)^2]
Distance = √[144 + 25]
Distance = √169
Distance = 13
Step 3: Find the sum of the radii
Sum of radii = √22 + 3
Sum of radii = √22 + √9
Sum of radii = √22 + 3
Step 4: Calculate the shortest distance
Shortest distance = Distance between centers - Sum of radii
Shortest distance = 13 - (√22 + 3)
Therefore, the shortest distance between the circles is 13 - (√22 + 3).