The points of intersection of 2 equal circles which cut orthogonal (2, 3 ) (5, 4) then the radius of each a circle is?

Question

Answer

Problem

Find the radius of two equal circles that intersect at two points (2, 3) and (5, 4) and cut orthogonally.

Solution

Let the centers of two equal circles be (x1, y1) and (x2, y2) and the radius be r.

Step 1: Find the distance between the centers

Using the distance formula, we can find the distance between the centers of the circles.

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Let the distance between the centers be d.

Step 2: Find the equation of the line passing through the centers

Since the circles cut orthogonally, the line passing through the centers of the circles is perpendicular to the line joining the two points of intersection.

Let the equation of the line passing through the centers be ax + by + c = 0.

Step 3: Find the equations of the circles

Using the equation of the line passing through the centers and the distance between the centers, we can find the equations of the circles.

x^2 + y^2 + 2gx + 2fy + c = 0

where g = (x1 + x2)/2, f = (y1 + y2)/2 and c = (x1 - x2)^2/4 + (y1 - y2)^2/4 - r^2

Step 4: Solve the equations of the circles

Solving the equations of the circles, we get:

x^2 + y^2 - 7x + 5y - 5 = 0

x^2 + y^2 - 3x - y - 3 = 0

Step 5: Find the radius of the circle

The radius of the circle can be found by using the equation:

r^2 = g^2 + f^2 - c

Substituting the values of g, f and c, we get:

r^2 = 29/2

Therefore, the radius of each circle is sqrt(29/2).

Answer

√5

Answer

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