Solutions

Algebra Level 3

Find the no.of solutions of the system of equations ∣ x ∣ + ∣ y ∣ = 1 \left| x \right| +\left| y \right| =1 and x 2 + y 2 = a 2 x^2+y^2=a^2 , 1 / 2 < a < 1 1/\sqrt { 2 } <a<1


The answer is 8.

Bala Vidyadharan by Bala vidyadharan , 20, India

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1 solution

Tom Engelsman
Sep 30, 2020

The circle x 2 + y 2 = 1 x^2 + y^2 = 1 circumscribes the unit square ∣ x ∣ + ∣ y ∣ = 1 |x|+|y| = 1 at the points ( ± 1 , 0 ) ; ( 0 , ± 1 ) (\pm1, 0); (0, \pm1) . The circle x 2 + y 2 = 1 2 x^2 + y^2 = \frac{1}{2} is internally tangent to the same unit square at the points ( ± 1 / 2 , 1 / 2 ) ; ( ± 1 / 2 , − 1 / 2 ) . (\pm 1/2, 1/2); (\pm 1/2, -1/2). Hence, there are 8 \boxed{8} total solutions to this system.

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