Explanation Given: The largest circle has a radius 2 and the 2 inscribed circles have radius 1 and the smallest circle is tangent to the largest circle and small circle and touches the x axis. Formula used: Pythagoras theorem: For a right-angled triangle , the square of length of hypotenuse is equal to the sum of squares of its perpendicular and base, i.e., H 2 = P 2 + B 2 where H is the hypotenuse P is the perpendicular and B is the base. The equation of a circle is given as ( x − a ) 2 + ( y − b ) 2 = r 2 , where r is the radius of the circle, a and b are the x and y intercepts of the center of the circle. For two numbers a and b , ( a + b ) 2 = a 2 + b 2 + 2 a b ( a − b ) 2 = a 2 + b 2 − 2 a b Calculation: Let the radius of the required circle be r and the co-ordinates of its center be C ( h , k ) . Mark certain points as shown in diagram and draw the perpendicular from the center to the co-ordinate axes and join the center to origin and center of the circle with radius 1. As the circle touches the x axis, the y-co-ordinate is equal to negative of radius. Now, apply Pythagoras theorem for triangle A O C , ( 2 − r ) 2 = ( − r ) 2 + h 2 Hence, h 2 = ( 2 − r ) 2 − r 2 …… (1) Similarly, apply Pythagoras theorem for triangle B O C . ( r + 1 ) 2 = ( 1 − r ) 2 + h 2 Hence, h 2 = ( r + 1 ) 2 − ( r − 1 ) 2 …… (2) Substitute value of h 2 from equation (1) and equation (2), ( 2 − r ) 2 − r 2 = ( r + 1 )

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ISBN: 9780321916600

Author: Mark Dugopolski

Publisher: PEARSON

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Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 5.3, Problem 85E

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To calculate: The equation of the smallest circle, where the largest circle has a radius 2 and the 2 inscribed circles have radius 1 and the smallest circle is tangent to the largest circle and a small which touches the three larger circles and the required circle, and the required circle touches the x axis.

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Chapter 5.3, Problem 84E

Chapter 5.3, Problem 86E

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College Algebra (6th Edition)

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The equation of the smallest circle, where the largest circle has a radius 2 and the 2 inscribed circles have radius 1 and the smallest circle is tangent to the largest circle and a small which touches the three larger circles and the required circle, and the required circle touches the x axis.

Solution Summary: The author calculates the equation of the smallest circle, where the largest circle has a radius 2 and the 2 inscribed circles have radius 1.

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