Moving along a Circle

Geometry Level 2

For point ( x , y ) (x,y) moving along the circle x 2 + y 2 = 4 x^2+y^2=4 , let a a and b b be the maximum and minimum values, respectively, of 1 1 + x 2 + 1 1 + y 2 . \frac{1}{1+x^2}+\frac{1}{1+y^2}. What is the value of a b ab ?

6 5 \frac{6}{5} 4 5 \frac{4}{5} 3 5 \frac{3}{5} 1

Ajay Dutta by Ajay Dutta , 24, India

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1 solution

Arjun Nemani
Mar 3, 2014

6 5 + x 2 y 2 \frac { 6 }{ 5+x^2y^ 2 } Max Value would be when denominator is min. which will be when X^2Y^2, will be min : which is "0".. Min Value can be thought via trig no metry and writing the equation of the circle in parametric form and finding the max of the x^2y^2 term. (Hint : 45Deg)

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