\n
Enunciado<\/b><\/span><\/p>\nConsidere os n\u00fameros complexos:<\/p>\n
$$\\begin{array}{*{20}{l}}
\n{z = x + yi\\,\\,{\\text{de afixo M}}}&;&{{z_1} = x – 4 + i\\left( {y + 5} \\right)}&{\\text{e}}&{{z_2} = x + 4 + i\\left( {1 – y} \\right)}
\n\\end{array}$$<\/p>\n
\n- Para que valores de $x$ e $y$ se tem ${z_1} = 3{z_2}$?<\/li>\n
- Determine e represente no plano complexo o conjunto $C_1$ dos pontos M tais que ${z_1} + {z_2}$ seja um imagin\u00e1rio puro.<\/li>\n
- Determine e represente no plano complexo o conjunto $C_2$ dos pontos M tais que ${z_1}.{z_2}$ seja real.<\/li>\n
- Mostre que as duas\u00a0condi\u00e7\u00f5es seguintes s\u00e3o equivalentes:\n
a) ${x^2} + {\\left( {y + 2} \\right)^2} = 25$<\/p>\n
b) ${z_1}.{z_2}$\u00a0\u00e9 um imagin\u00e1rio puro<\/p>\n<\/li>\n
- Seja A o afixo de $ – 2i$.
\nDeduza que o conjunto $C_3$ dos pontos M tais que ${z_1}.{z_2}$\u00a0\u00e9 um imagin\u00e1rio puro \u00e9 a circunfer\u00eancia de centro A e raio 5.<\/li>\n<\/ol>\nResolu\u00e7\u00e3o >><\/a><\/span><\/div><\/div>\n\n\n
Resolu\u00e7\u00e3o<\/b><\/span><\/p>\n\n- Ora,
\n$$\\begin{array}{*{20}{l}}
\n{{z_1} = 3{z_2}}& \\Leftrightarrow &{x – 4 + i\\left( {y + 5} \\right) = 3\\left( {x + 4 + i\\left( {1 – y} \\right)} \\right)} \\\\
\n{}& \\Leftrightarrow &{\\begin{array}{*{20}{l}}
\n{x – 4 = 3x + 12}& \\wedge &{y + 5 = 3 – 3y}
\n\\end{array}} \\\\
\n{}& \\Leftrightarrow &{\\begin{array}{*{20}{l}}
\n{x =\u00a0 – 8}& \\wedge &{y =\u00a0 – \\frac{1}{2}}
\n\\end{array}}
\n\\end{array}$$
\nPortanto, ${z_1} = 3{z_2}$ para ${\\begin{array}{*{20}{l}}
\n{x =\u00a0 – 8}& \\wedge &{y =\u00a0 – \\frac{1}{2}}
\n\\end{array}}$.
\n\u00ad<\/li>\n - Como $$\\begin{array}{*{20}{l}}
\n{{z_1} + {z_2}}& = &{x – 4 + i\\left( {y + 5} \\right) + x + 4 + i\\left( {1 – y} \\right)} \\\\
\n{}& = &{2x + 6i}
\n\\end{array}$$ ent\u00e3o $$\\begin{array}{*{20}{l}}
\n{{z_1} + {z_2}{\\text{ \u00e9\u00a0 imagin\u00e1rio puro}}}& \\Leftrightarrow &{2x = 0} \\\\
\n{}& \\Leftrightarrow &{x = 0}
\n\\end{array}$$
\nLogo, ${C_1} = \\left\\{ {\\left( {x,y} \\right) \\in {\\mathbb{R}^2}:x = 0 \\wedge y \\in \\mathbb{R}} \\right\\}$.
\n\u00ad<\/li>\n - Como $$\\begin{array}{*{20}{l}}
\n{{z_1}.{z_2}}& = &{\\left( {x – 4 + i\\left( {y + 5} \\right)} \\right).\\left( {x + 4 + i\\left( {1 – y} \\right)} \\right)} \\\\
\n{}& = &{{x^2} – 16 + i\\left( {x – 4 – xy + 4y} \\right) + i\\left( {xy + 4y + 5x + 20} \\right)} \\\\
\n{}& = &{{x^2} + {y^2} + 4y – 21 + i\\left( {6x + 8y + 16} \\right)}
\n\\end{array} – \\left( {y – {y^2} + 5 – 5y} \\right)$$ ent\u00e3o $$\\begin{array}{*{20}{l}}
\n{{z_1}.{z_2}{\\text{ \u00e9\u00a0 real}}}& \\Leftrightarrow &{6x + 8y + 16 = 0} \\\\
\n{}& \\Leftrightarrow &{y =\u00a0 – \\frac{3}{4}x – 2}
\n\\end{array}$$
\nLogo, ${C_2} = \\left\\{ {\\left( {x,y} \\right) \\in {\\mathbb{R}^2}:y =\u00a0 – \\frac{3}{4}x – 2} \\right\\}$.
\n\u00ad<\/li>\n - De facto, $$\\begin{array}{*{20}{l}}
\n{{z_1}.{z_2}{\\text{ \u00e9\u00a0 imagin\u00e1rio puro}}}& \\Leftrightarrow &{{x^2} + {y^2} + 4y – 21 = 0} \\\\
\n{}& \\Leftrightarrow &{{x^2} + {{\\left( {y + 2} \\right)}^2} – 4 – 21 = 0} \\\\
\n{}& \\Leftrightarrow &{{x^2} + {{\\left( {y + 2} \\right)}^2} = 25}
\n\\end{array}$$
\n\u00ad<\/li>\n - De facto, ${C_3} = \\left\\{ {\\left( {x,y} \\right) \\in {\\mathbb{R}^2}:{x^2} + {{\\left( {y + 2} \\right)}^2} = 25} \\right\\}$ \u00e9 a circunfer\u00eancia de centro A e raio 5 unidades, lugar geom\u00e9trico dos pontos M tais que ${z_1}.{z_2}$\u00a0\u00e9 um imagin\u00e1rio puro.
\n\u00ad<\/li>\n<\/ol>\n