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1. In the coordinate plane, let A be a point on the x-axis, and let B be a point on the y-axis, so that AB is tangent to the unit circle. Find the minimum value of AB. 2. Let a, b, c, d, e, f, g, and h be positive real numbers such that abcd = 4 and efgh = 9. Find the minimum value of May 30, 2020

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1. The minimum value is sqrt(5).

2. The minimum value is 36.

May 30, 2020
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1.  From symmetry, the minimum occurs when OA = OB, where O is the origin.  The tangent line (touching the circle at T, say)  is then at an angle of 45degrees to the horizontal, and 90 degrees to the radial line OT, so forms 45/45/90 right angled triangles with horizontal and vertical axes. This means TA and TB are the same length as OT, which makes AB = 2

2. Again, symmetry dictates that there is nothing to distinguish a, b, c and d from each other, nor e, f, g and h from each other, so we have :

\(a^4=4\\e^4=9\)

and \((ae)^2+(bf)^2+(cg)^2+(dh)^2=4.a^2.e^2=4.2.3=24\)

Jun 2, 2020
edited by Alan  Jun 2, 2020