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Question Number 218526 by Spillover last updated on 11/Apr/25

Commented by mr W last updated on 11/Apr/25

$$\approx 0.8348$$

Commented by Spillover last updated on 11/Apr/25

$$correct$$

Answered by mr W last updated on 11/Apr/25

Commented by mr W last updated on 12/Apr/25

$$‘‘a^{″}inoriginalquestionisreplaced$$$$with2sinfollowing.$$$$sayA(a,0),B(0,b)$$$$r=\frac{ab}{a+b+\sqrt{a^2+b^2}}$$$$\frac{x}{a}+\frac{y}{b}=1orbx+ay=ab\leftarrow eqn.ofAB$$$$s=\frac{bs+as-ab}{\sqrt{a^2+b^2}}$$$$(a+b-\sqrt{a^2+b^2})s=ab$$$$say\alpha =\frac{a}{s},\beta =\frac{b}{s}$$$$\alpha +\beta -\sqrt{{\alpha }^2+{\beta }^2}=\alpha \beta$$$$\beta =\frac{2(1-\alpha )}{2-\alpha }...\left(I\right)$$$$$$$${(x-a)}^2+y^2=a^2+b^2$$$$x^2-2ax+y^2=b^2...\left(i\right)$$$${(x-s)}^2+{(y-s)}^2=s^2$$$$x^2-2sx+y^2-2sy=-s^2...\left(ii\right)$$$$\left(i\right)-\left(ii\right):$$$$2(s-a)x+2sy=b^2+s^2\leftarrow eqn.ofintersectionline$$$$d=\frac{2(s-a)s+2s^2-b^2-s^2}{2\sqrt{{(s-a)}^2+s^2}}=s-2r$$$$\frac{3s^2-2as-b^2}{2\sqrt{2s^2-2sa+a^2}}=s-\frac{2ab}{a+b+\sqrt{a^2+b^2}}$$$$\frac{3-2\alpha -{\beta }^2}{2\sqrt{2-2\alpha +{\alpha }^2}}=1-\frac{2\alpha \beta }{\alpha +\beta +\sqrt{{\alpha }^2+{\beta }^2}}...\left(II\right)$$$$$$$$from\left(I\right)and\left(II\right)weget$$$$\alpha \approx 0.677564399386,\beta \approx 0.487639020704$$$$\frac{2AB}{‘‘a^{″}}=\frac{AB}{s}=\frac{\sqrt{a^2+b^2}}{s}$$$$=\sqrt{{\alpha }^2+{\beta }^2}\approx 0.834796579910✓$$

Commented by mr W last updated on 11/Apr/25

Commented by Spillover last updated on 17/Apr/25

$$GODblessyou.very?nice$$$$solution$$

Answered by Spillover last updated on 11/Apr/25

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