1. two circles of radii r₁,r₂ cuts orthoganali then area included between the circles is

When two circles intersect orthogonally, it means they cut each other at right angles. This unique relationship between the circles allows us to calculate the area enclosed between them. Let’s break down the concept and the calculation step by step.

Understanding Orthogonal Circles

Two circles are said to intersect orthogonally if the tangents to each circle at the points of intersection are perpendicular to each other. Mathematically, this condition can be expressed as:

• If the centers of the circles are at points \(C_1\) and \(C_2\), and their respective radii are \(r_1\) and \(r_2\), then the circles intersect orthogonally if:
• \(d^2 = r_1^2 + r_2^2\)

Here, \(d\) is the distance between the centers of the circles. This relationship is crucial for determining the area between the circles.

Calculating the Area Between the Circles

The area enclosed between two orthogonal circles can be derived using the formula:

Area = π * (r₁² + r₂²)

This formula arises from the fact that the area of each circle contributes to the total area, but since they intersect, we need to account for the overlapping region only once.

Step-by-Step Derivation

Let’s delve deeper into how we arrive at this formula:

1. Calculate the area of each circle: The area of a circle is given by the formula \(A = πr²\). Therefore, the areas of the two circles are:
• Area of Circle 1: \(A_1 = πr_1²\)
• Area of Circle 2: \(A_2 = πr_2²\)
2. Combine the areas: Since the circles intersect, we need to add their areas together:
• Total Area = \(A_1 + A_2 = πr_1² + πr_2²\)
3. Adjust for the intersection: The area of the intersection is not directly subtracted because we are interested in the area that is uniquely between the circles. The orthogonal condition simplifies this to:
• Area Between = \(π(r_1² + r_2²)\)

Example for Clarity

Let’s say we have two circles with radii \(r_1 = 3\) and \(r_2 = 4\). To find the area between them:

• Calculate the area of Circle 1: \(A_1 = π(3²) = 9π\)
• Calculate the area of Circle 2: \(A_2 = π(4²) = 16π\)
• Combine the areas: \(Total Area = 9π + 16π = 25π\)
• Thus, the area between the circles is: \(Area = π(3² + 4²) = π(9 + 16) = 25π\)

Final Thoughts

In summary, when two circles intersect orthogonally, the area between them can be calculated using the formula \(π(r_1² + r_2²)\). This approach not only highlights the beauty of geometry but also emphasizes the importance of understanding the relationships between different shapes. If you have any further questions or need clarification on any part of this topic, feel free to ask!

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