Question

Considerando um plano de coordenadas cartesianas, o comprimento da corda que a reta r de equação 2x−y+1=0 forma com a elipse $\alpha$ :x2+y22=94 mede, em unidades de comprimento:

Answer 1

Para determinar o comprimento da corda em relação à elipse, primeiro temos que encontrar os pontos onde a reta e a elipse se interceptam e assim fazer a distância entre esses dois pontos.

Temos :

$\displaystyle \text{reta} : 2\text x - \text y + 1 =0 \\\\ \text{elipse} : \text x^2+\frac{\text y^2}{2} = \frac{9}{4}$

É um sistema de duas equação e duas incógnitas, resolvendo

$\displaystyle \text y = 2\text x+1 \\\\ \text x^2+\frac{(2\text x+1)^2}{2}=\frac{9}{4} \\\\\\ \frac{2\text x^2 +4\text x^2+4\text x+1}{2}=\frac{9}{4} \\\\\\ 6\text x^2+4\text x+1 = \frac{9}{2} \\\\ 12\text x^2+8\text x+2-9 = 0 \\\\ 12\text x^2+8\text x-7=0 \\\\ \text x = \frac{-8\pm\sqrt{8^2-4.12.(-7)}}{2.12} \\\\\\ \text x = \frac{-8\pm\sqrt{64+336}}{2.12} \\\\\\ \text x = \frac{-8\pm20 }{2.12}$

$\displaystyle \text x = \frac{-8-20}{2.12} \to \boxed{\text x = \frac{-7}{6}}\\\\ \text x= \frac{-8+20}{2.12} \to \boxed{\text x = \frac{1}{2}}$

Achando os valores de y :

*

$\displaystyle \text y = 2\text x+1 \to \text x = \frac{1}{2} \to \text y = 2\\\\ \underline{\text{ponto 1}}:(\frac{1}{2},2)$

*

$\displaystyle \text y = 2\text x+1 \to \text x = \frac{-7}{6} \to \text y = \frac{2(-7)}{6}+1 \\\\ \text y = \frac{-7+3}{3} \\\\ \underline{\text{ponto 2}}:(\frac{-7}{6},\frac{-4}{3})$

Fazendo a distância entre o ponto 1 e 2 :

$\displaystyle \text D = \sqrt{(\frac{1}{2}-(\frac{-7}{6}))^2+(2-(\frac{-4}{3}))^2} \\\\\\ \text D = \sqrt{(\frac{1}{2}+\frac{7}{6})^2+(2+\frac{4}{3})^2} \\\\\\ \text D =\sqrt{(\frac{3}{6}+\frac{7}{6})^2+(\frac{6+4}{3})^2} \\\\\\ \text D =\sqrt{\frac{100}{36}+\frac{100.4}{9.4}} \\\\\\ \text D = \sqrt{\frac{500}{36}} \to \text D =\sqrt{\frac{5.10^2}{6^2}} \\\\\\ \text D = \frac{10\sqrt{5}}{6} \\\\\\ \huge\boxed{\text D = \frac{5\sqrt5}{3}}\checkmark$

Letra B