\(a,\Leftrightarrow\left(x^2-2xy+y^2\right)+\left(x^2+x+\dfrac{1}{4}\right)+\dfrac{7}{4}=0\\ \Leftrightarrow\left(x-y\right)^2+\left(x+\dfrac{1}{2}\right)^2+\dfrac{7}{4}=0\\ \Leftrightarrow x,y\in\varnothing\left[\left(x-y\right)^2+\left(x+\dfrac{1}{2}\right)^2+\dfrac{7}{4}\ge\dfrac{7}{4}>0\right]\\ b,\Leftrightarrow\left(x^2-2x+1\right)+\left(9y^2+12y+4\right)+\left(4z^2-4z+1\right)+14=0\\ \Leftrightarrow\left(x-1\right)^2+\left(3y+2\right)^2+\left(2z-1\right)^2+14=0\\ \Leftrightarrow x,y,z\in\varnothing\left[\left(x-1\right)^2+\left(3y+2\right)^2+\left(2z-1\right)^2+14\ge14>0\right]\)

\(c,\Leftrightarrow-\left(x^2-10xy+25y^2\right)-\left(y^2-20y+100\right)-50=0\\ \Leftrightarrow-\left(x-5y\right)^2-\left(y-10\right)^2-50=0\\ \Leftrightarrow x,y\in\varnothing\left[-\left(x-5y\right)^2-\left(y-10\right)^2-50\le-50< 0\right]\)

\(x^2+2x+y^2-6y+4z^2-4z+11=0\)

\(\Leftrightarrow x^2+2x+1+y^2-6y+9+4z^2-4z+1=0\)

\(\Leftrightarrow\left(x+1\right)^2+\left(y-3\right)^2+\left(2z-1\right)^2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}x+1=0\\y-3=0\\2z-1=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=-1\\y=3\\z=\dfrac{1}{2}\end{matrix}\right.\)

\(x^2+2x+y^2-6y+4z^2-4z+11=0\\ \Rightarrow\left(x^2+2x+1\right)+\left(y^2-6y+9\right)+\left(4z^2-4z+1\right)=0\\ \Rightarrow\left(x+1\right)^2+\left(y-3\right)^2+\left(2z-1\right)^2=0\)

Vì \(\left(x+1\right)^2\ge0;\left(y-3\right)^2\ge0;\left(2z-1\right)^2\ge0\) mà \(\left(x+1\right)^2+\left(y-3\right)^2+\left(2z-1\right)^2=0\)

\(\Rightarrow\left\{{}\begin{matrix}\left(x+1\right)^2=0\\\left(y-3\right)^2=0\\\left(2z-1\right)^2=0\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}x=-1\\y=3\\z=\dfrac{1}{2}\end{matrix}\right.\)

Bài làm:

Ta có: \(x^2+2x+y^2-6y+4z^2-4z+11=0\)

\(\Leftrightarrow\left(x^2+2x+1\right)+\left(y^2-6y+9\right)+\left(4z^2-4z+1\right)=0\)

\(\Leftrightarrow\left(x+1\right)^2+\left(y-3\right)^2+\left(2z-1\right)^2=0\)

Dấu "=" xảy ra khi: \(\hept{\begin{cases}\left(x+1\right)^2=0\\\left(y-3\right)^2=0\\\left(2z-1\right)^2=0\end{cases}}\Rightarrow\hept{\begin{cases}x=-1\\y=3\\x=\frac{1}{2}\end{cases}}\)

Xin lỗi mk nhầm đoạn cuối là: \(\Rightarrow\hept{\begin{cases}x=-1\\y=3\\z=\frac{1}{2}\end{cases}}\) nhé:)

<=>x²+2x+1+y²-6y+9+4z²-4z+1=0

<=>(x²+2x+1)+(y²-6y+9)+(4z²-4z+1)=0

<=>(x+1)²+(y-3)²+(2z+1)²=0       (1)

Từ (1) <=> (x+1)²=0 <=> x+1=0 <=>x=-1

                  (y-3)²=0 <=>y-3=0 <=>y=3

                  (2z+1)²=0 <=>2z+1=0 <=> z=-1/2

x²+2x+y²-6y+4z^2-4z+11=0

\(\Leftrightarrow\left(x^2+2x+1\right)+\left(y^2-6y+9\right)+\left(4z^2-4z+1\right)=0\)

<=>(x+1)²+(y-3)²+(2z-1)²=0

Vì (x+1)^2\(\ge\)0;(y-3)^2\(\ge\)0;(2z-1)²\(\ge\)0 => (x+1)²+(y-3)²+(2z-1)²\(\ge\)0

Dấu "=" xảy ra khi (x+1)²=(y-3)²=(2z-1)²=0 <=> x+1=y-3=2z-1=0 <=> x=-1;y=3;z=1/2

\(\Leftrightarrow\hept{\begin{cases}6x-5y=0\\8y-4z=0\\2x+y-z-4=0\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}6x=5y\\2y=z\\2x+y-z=4\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}\frac{x}{5}=\frac{y}{6}=\frac{z}{12}\\2x+y-z=4\end{cases}}\)

\(\Leftrightarrow\frac{x}{5}=\frac{y}{6}=\frac{z}{12}=\frac{2x+y-z}{10+6-12}=\frac{4}{4}=1\)

\(\Rightarrow x=5\)

      \(y=6\)

       \(z=12\)

Gọi mặt phẳng là (P) dễ kí hiệu

\(d\left(M;\left(P\right)\right)=\frac{\left|-6+2+2-7\right|}{\sqrt{2^2+2^2+1}}=\frac{9}{3}=3\)

Áp dụng định lý Pitago:

\(R=\sqrt{3^2+4^2}=5\)

Phương trình mặt cầu:

\(\left(x+3\right)^2+\left(y-1\right)^2+\left(z-2\right)^2=25\)

\(\Leftrightarrow x^2+y^2+z^2+6x-2y-4z-11=0\)

\(a,9x^2+y^2+2z^2-18x+4z-6y+20=0\\ \Leftrightarrow9\left(x-1\right)^2+\left(y-3\right)^2+2\left(z+1\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}x=1\\y=3\\z=-1\end{matrix}\right.\)

\(b,5x^2+5y^2+8xy+2y-2x+2=0\\ \Leftrightarrow4\left(x+y\right)^2+\left(x-1\right)^2+\left(y+1\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}x=-y\\x=1\\y=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=-1\end{matrix}\right.\)

\(c,5x^2+2y^2+4xy-2x+4y+5=0\\ \Leftrightarrow\left(2x+y\right)^2+\left(x-1\right)^2+\left(y+2\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}2x=-y\\x=1\\y=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=-2\end{matrix}\right.\)

\(d,x^2+4y^2+z^2=2x+12y-4z-14\\ \Leftrightarrow\left(x-1\right)^2+\left(2y-3\right)^2+\left(z+2\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}x=1\\y=\dfrac{3}{2}\\z=-2\end{matrix}\right.\)

\(e,x^2+y^2-6x+4y+2=0\\ \Leftrightarrow\left(x-3\right)^2+\left(y+2\right)^2=11\)

Pt vô nghiệm do ko có 2 bình phương số nguyên có tổng là 11

e: Ta có: \(x^2-6x+y^2+4y+2=0\)

\(\Leftrightarrow x^2-6x+9+y^2+4y+4-11=0\)

\(\Leftrightarrow\left(x-3\right)^2+\left(y+2\right)^2=11\)

Dấu '=' xảy ra khi x=3 và y=-2