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Question Number 189684 by normans last updated on 20/Mar/23

Commented by a.lgnaoui last updated on 21/Mar/23

Answered by mr W last updated on 20/Mar/23

Commented by mr W last updated on 21/Mar/23

$$x^2+y^2={\left(\sqrt{5}a\right)}^2=5a^2$$$${(16-y-y)}^2+{22}^2={\left(2\sqrt{5}a\right)}^2$$$$y^2-16y+185-5a^2=0...\left(i\right)$$$$\Rightarrow y=8-\sqrt{5a^2-121}$$$${(22-x-x)}^2+{16}^2={\left(4a\right)}^2$$$$x^2-22x+185-4a^2=0...\left(ii\right)$$$$\Rightarrow x=11-\sqrt{4a^2-64}$$$$\left(i\right)+\left(ii\right):$$$$11x+8y=185-2a^2$$$$11(11-\sqrt{4a^2-64})+8(8-\sqrt{5a^2-121})=185-2a^2$$$$11\sqrt{4a^2-64}=2a^2-8\sqrt{5a^2-121}$$$$a^2-41=8\sqrt{5a^2-121}$$$$a^4-402a^2+10705=0$$$$\Rightarrow a^2=201-32\sqrt{29}\approx 28.675$$

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