1. Gọi I chính là giao điểm của BD và AC. Ta có: AB = BC = DC = AD = AH + BH = 7+2 = 9(cm)

Xét\(\Delta AHD\left(\widehat{AHD}=90^0\right)\) theo định lý py - ta - go ta có : 

\(HD=\sqrt{AD^2-AH^2}=\sqrt{9^2-7^2}=4\sqrt{2}cm\)

Xét\(\Delta BHD\left(\widehat{BHD=90^O}\right)\)theo định lý py - ta - go ta có : 

\(BD=\sqrt{HD^2+BH^2}=\sqrt{\left(4\sqrt{2}\right)^2+2^2}=6cm\)

BI = DI =\(\frac{BD}{2}=\frac{6}{2}=3cm\). Xét\(\Delta AID\left(\widehat{AID}=90^O\right)\)theo định lý py - ta - go ta có : 

\(AI=\sqrt{AD^2-DI^2}=\sqrt{9^2-3^2}=6\sqrt{2cm}\)

AC = AI.2 =\(6\sqrt{2}.2=12\sqrt{2}\)=> S_ABCD =\(\frac{1}{2}.\left(BD.AC\right)=\frac{1}{2}.\left(6.12\sqrt{2}\right)=36\sqrt{2}cm\)

+) \(P=\frac{x}{\sqrt{1-x^2}}+\frac{y}{\sqrt{1-y^2}}=\frac{x^2}{x\sqrt{1-x^2}}+\frac{y^2}{y\sqrt{1-y^2}}\)

\(\ge\frac{\left(x+y\right)^2}{x\sqrt{1-x^2}+y\sqrt{1-y^2}}=\frac{1}{x\sqrt{1-x^2}+y\sqrt{1-y^2}}\)

+) \(A=x\sqrt{1-x^2}+y\sqrt{1-y^2}\)

\(A^2=x^2+y^2-y^4-x^4+2xy\sqrt{\left(1-x^2\right)\left(1-y^2\right)}\)

+) \(B=x^2+y^2-x^4-y^4=x^2+\left(1-x\right)^2-x^4-\left(1-x\right)^4\)

\(-\frac{B}{2}+\frac{3}{16}=x^4-2x^3+2x^2-x+\frac{3}{16}=\left(x^2-x+\frac{3}{4}\right)\left(x-\frac{1}{2}\right)^2\ge0\)

\(\Leftrightarrow B\le\frac{3}{8}\)

+) \(A^2\le\frac{3}{8}+2\frac{\left(x+y\right)^2}{4}\sqrt{1-x^2-y^2+x^2y^2}\)

\(\le\frac{3}{8}+\frac{1}{2}\sqrt{1-\frac{\left(x+y\right)^2}{2}+\frac{\left(x+y\right)^4}{16}}=\frac{3}{8}+\frac{1}{2}\sqrt{1-\frac{1}{2}+\frac{1}{16}}=\frac{3}{8}+\frac{1}{2}\cdot\frac{3}{4}=\frac{3}{4}\)

\(\Rightarrow A\le\frac{\sqrt{3}}{2}\)

+) \(P=\frac{1}{A}\ge\frac{2\sqrt{3}}{3}\)

Vậy \(P_{min}=\frac{2\sqrt{3}}{3}\)khi \(x=y=\frac{1}{2}\)

* Mình làm hơi tắt và có vẻ hơi dài

Từ điều kiện đề bài ta có: \(P=\frac{x}{\sqrt{y^2+2xy}}+\frac{y}{\sqrt{x^2+2xy}}\)

Theo Holder: \(P.P.\left[x\left(y^2+2xy\right)+y\left(x^2+2xy\right)\right]\ge\left(x+y\right)^3\)

\(\Rightarrow P^2\ge\frac{\left(x+y\right)^3}{x\left(y^2+2xy\right)+y\left(x^2+2xy\right)}\) (*)

Xét: \(\frac{\left(x+y\right)^3}{x\left(y^2+2xy\right)+y\left(x^2+2xy\right)}-\frac{4}{3}=\frac{\left(x+y\right)\left(x-y\right)^2}{x\left(y^2+2xy\right)+y\left(x^2+2xy\right)}\ge0\) (**)

Từ (*) và (**) suy ra: \(P\ge\frac{2}{\sqrt{3}}\)

Dấu "=" xảy ra khi x=y=1\2

=\(\sqrt{\left(5+2\sqrt{6}\right)+\left(2\sqrt{10}+2\sqrt{15}\right)+5}\)

=\(\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2+2\sqrt{5}\left(\sqrt{2}+\sqrt{3}\right)+\left(\sqrt{5}\right)^2}\)

=\(\sqrt{\left(\sqrt{3}+\sqrt{2}+\sqrt{5}\right)^2}\)

=\(\sqrt{3}+\sqrt{2}+\sqrt{5}\)

\(P=\frac{\sqrt{1+x^2+y^2}}{xy}+\frac{\sqrt{1+y^2+z^2}}{yz}+\frac{\sqrt{1+z^2+x^2}}{zx}\)

\(\ge\text{Σ}\frac{\sqrt{\frac{\left(1+x+y\right)^2}{3}}}{xy}\text{=}\frac{1+x+y}{xy\sqrt{3}}\)

\(=\frac{\sqrt{3}}{3}\left(\frac{1+x+y}{xy}+\frac{1+y+z}{yz}+\frac{1+z+x}{zx}\right)\)

\(=\frac{\sqrt{3}}{3}\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz}+\frac{1}{x}+\frac{1}{y}+\frac{1}{y}+\frac{1}{z}+\frac{1}{z}+\frac{1}{x}\right)\)

\(=\frac{\sqrt{3}}{3}\left(x+y+z+2xy+2yz+2zx\right)\)\(\ge\frac{\sqrt{3}}{3}\left(3\sqrt[3]{xyz}+2\cdot3\sqrt[3]{x^2y^2z^2}\right)=\frac{\sqrt{3}}{3}\left(3+6\right)=3\sqrt{3}\)

Dấu = xảy ra khi \(x=y=z=1\)

b) Đk: \(0\le x\le4\)

Ta có: \(\sqrt{4x+x^2}+\sqrt{4x-x^2}=4x+1\)

<=> \(\left(\sqrt{4x+x^2}+\sqrt{4x-x^2}\right)^2=\left(4x+1\right)^2\)

<=> \(\left|4x+x^2\right|+\left|4x-x^2\right|+2\sqrt{\left(4x+x^2\right)\left(4x-x^2\right)}=16x^2+8x+1\)

<=> \(x^2+4x+4x-x^2+2x\sqrt{\left(4-x\right)\left(4+x\right)}=16x^2+8x+1\)

<=> \(2x\sqrt{16-x^2}=16x^2+8x+1-8x\)

<=> \(\left(2x\sqrt{16-x^2}\right)^2=\left(16x^2+1\right)^2\)

<=> \(4x^2\left|16-x^2\right|=256x^4+32x^2+1\)

<=> \(64x^2-4x^4=256x^4+32x^2+1\)

<=> \(260x^4-32x^2+1=0\)

Đặt x² = k (k > 0) <=> 260k² - 32k + 1 = 0

Ta có: \(\Delta=32^2-4.260=-16< 0\)

=> pt vô nghiệm

\(\sqrt{4x+x^2}+\sqrt{4x-x^2}=4x+1\) đk: \(0\le x\le4\)

\(\Leftrightarrow4x+x^2+4x-x^2+2\sqrt{16x^2-x^4}=16x^2+8x+1\)

\(2\sqrt{16x^2-x^4}=16x^2+1\)

\(\Leftrightarrow64x^2-4x^4=256x^4+32x^2+1\)

\(\Leftrightarrow260x^2-32x^2+1=0\)

=> Vo nghiem

+) \(\left(x^2+1\right)\left(x+1\right)=4^y\Leftrightarrow\left(x^2+1\right)\left(x+1\right)=2^{2y}\)

+) Do  \(x,y\inℕ\)nên ta có \(x^2+1=2^m\)và \(x+1=2^n\)với \(m+n=2y;m,n\inℕ\)

+) Lúc đó ta có: \(\orbr{\begin{cases}x^2+1⋮x+1\\x+1⋮x^2+1\end{cases}}\)

TH1: \(x^2+1⋮x+1\Leftrightarrow\left(x+1\right)^2-2\left(x+1\right)+2⋮x+1\)

\(\Leftrightarrow2⋮x+1\Leftrightarrow x\in\left\{0;1\right\}\)

TH2: \(x+1⋮x^2+1\Leftrightarrow x^2-1⋮x^2+1\Leftrightarrow2⋮x+1\)

\(\Leftrightarrow x\in\left\{0;1\right\}\)

* Nếu x = 0 thì \(4^y=1\Leftrightarrow y=0\)

* Nếu y = 0 thì \(4^y=4\Leftrightarrow y=1\)

Vậy \(\left(x;y\right)\in\left\{\left(0;0\right);\left(1;1\right)\right\}\)