The points of intersection of two equal circles which cut orthogonally are (2,3) and (5,4). Then radius of each circle is?
what a lot of bother I went to above.
The distance from (2,3) to (5,4) is √10.
Since the circles are orthogonal at the intersections, the two given points and the centers of the circles form a square, with diagonal √10.
So, the side of the square, which is the radius of the circles, is √5.
Well, it seems like those circles really hit it off and formed a love connection! They intersect at (2,3) and (5,4). Now, to find the radius of each circle, we'll need to do a little bit of math.
The distance between the two intersection points will give us the diameter of the circles. Using the distance formula (did you ever think you'd use that after high school?), we can find the diameter.
So, the distance between (2,3) and (5,4) is sqrt((5-2)^2 + (4-3)^2). Simplifying that gives us sqrt(9+1), which equals sqrt(10).
Now, since the diameter is twice the radius, we can divide the diameter (sqrt(10)) by 2 to find the radius.
Therefore, the radius of each circle is sqrt(10)/2. But hey, no need to be square-rootin' around all day - let's just call it approximately 1.58.
So, the radius of each circle is approximately 1.58 units.
To find the radius of each circle, we need to first find the center point of the circles.
Let's call the center point of the first circle (x₁, y₁) and the center point of the second circle (x₂, y₂).
Given that the circles cut orthogonally, the line connecting the centers of the circles passes through the points of intersection. So, we can find the equation of the line passing through (2, 3) and (5, 4) to find the center points.
First, let's find the slope of the line:
slope = (y₂ - y₁) / (x₂ - x₁)
slope = (4 - 3) / (5 - 2)
slope = 1 / 3
Now, using the point-slope form of the equation of a line:
y - y₁ = m(x - x₁)
y - 3 = (1/3)(x - 2)
y - 3 = (1/3)x - 2/3
Simplifying the equation, we get:
3y - 9 = x - 2
x - 3y = -7
This equation represents the line passing through the center points of the circles.
Now, let's find the midpoint of the line segment connecting the two points of intersection:
(x₁ + x₂) / 2 = (2 + 5) / 2 = 7 / 2 = 3.5
(y₁ + y₂) / 2 = (3 + 4) / 2 = 7 / 2 = 3.5
So, the midpoint of the line segment is (3.5, 3.5).
Since the line passing through the centers of the circles is perpendicular to the line connecting the points of intersection, their slopes are negative reciprocals.
The slope of the line passing through the center points is -(1/3).
So, using the equation y - y₁ = m(x - x₁), we can find the center point of the circles:
y - 3.5 = -(1/3)(x - 3.5)
y - 3.5 = -(1/3)x + (7/6)
3y - 10.5 = -x + (7/2)
Comparing the coefficients of x and y, we get:
-1 = -3y
7/2 = -10.5
Solving these equations, we get:
y = (7/6) and x = 3/2
Therefore, the center points of the circles are (3/2, 7/6).
Now, we can find the radius of each circle using the distance formula, considering one of the given points of intersection:
radius = √ [(x - x₁)² + (y - y₁)²]
radius = √ [(5 - 3/2)² + (4 - 7/6)²]
radius = √ [(10/2 - 3/2)² + (24/6 - 7/6)²]
radius = √ [(7/2)² + (17/6)²]
radius = √ [(49/4) + (289/36)]
radius = √ [(441/36) + (289/36)]
radius = √ (730/36)
radius = √ (365/18)
radius ≈ 5.35
Therefore, the radius of each circle is approximately 5.35.
To find the radius of each circle, we need to understand the properties of circles that cut orthogonally.
When two circles cut orthogonally, it means that the tangent at the point of intersection of the circles is perpendicular to each other. This indicates that the centers of both circles and the two given points of intersection form a rectangle.
Let's name the centers of the two circles as O1 and O2, and the radius as r. The coordinates of O1 and O2 can be found by finding the midpoint of the two given points of intersection.
Midpoint of (2,3) and (5,4):
x-coordinate of midpoint = (2 + 5)/2 = 3.5
y-coordinate of midpoint = (3 + 4)/2 = 3.5
So, the centers of both circles are O1(3.5, 3.5) and O2(3.5, 3.5).
Since the circles are equal, the distance of any of the given points of intersection from their respective centers will be equal to the radius.
Using the distance formula, the distance between (2,3) and (3.5, 3.5):
d = sqrt((2 - 3.5)^2 + (3 - 3.5)^2) = sqrt((-1.5)^2 + (-0.5)^2) = sqrt(2.25 + 0.25) = sqrt(2.5) ≈ 1.58
Therefore, the radius of each circle is approximately 1.58.
The line joining the two points is
y-3 = (1/3)(x-2)
Its midpoint is at (7/2,7/2)
The centers of the circles lie on its perpendicular bisector, which is
y-7/2 = -3(x-7/2)
If the radius of the circles is r, the distance from (2,3) to the second line is r/√2
So, just figure that distance, multiply by √2 and you have your radius.