A geometry problem by cool dude

Geometry Level 4

( tan ⁡ A − cos ⁡ B ) 2 + ( cot ⁡ A − sin ⁡ B ) 2 \large (\tan A - \cos B )^2 + ( \cot A - \sin B)^2

Let A A and B B be real numbers . If the minimum value of the expression can be expressed as b − a a b - a \sqrt a , where a a and b b are positive integers , find the value of a + b a+b .

5 7 2 10

Cool Dude by cool dude , 21, India

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2 solutions

Cool Dude
May 22, 2016

The given expression can be interpreted as square of distance between the points (tan A, cot A) and (cos B,sin B). (tan A,cot A) - Parametric coordinate of a point on the hyperbola xy = 1. (cos B,sin B) - Parametric coordinate of a point on the circle x^2 + y^2 = 1. The minimum distance between the curves xy = 1 and x^2 + y^2 = 1 can be calculated. Therefore, b - a(a)^0.5 = 3 - 2(2)^0.5, b = 3, a = 2, Therefore, a + b = 5

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Ayush G Rai
Jun 17, 2016

I substituted A and B as 45 and got the correct result

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