Question

A reta de equação y=2-x determina na circunferência x^+y^=4 uma corda.
a)Quanto vale o comprimento dessa corda?
b)Quais são as coordenadas do ponto médio dessa corda?

Answer 1

$\mathrm{y=2-x\ \to\ f(x)=2-x}\\ \mathrm{x^2+y^2=4\ \to\ y=\sqrt{4-x^2}\ \to\ g(x)=\sqrt{4-x^2}}\\\\ \textbf{Primeiramente, encontraremos os pontos de}\\ \textbf{intersec{c}\~ao entre as duas fun\c{c}\~oes:}\\\\ \mathrm{f(x)=g(x)\ \to\ 2-x=\sqrt{4-x^2}\ \to\ \big(2-x\big)^2=\big(\sqrt{4-x^2}\big)^2}\\ \mathrm{4-4x+x^2=4-x^2\ \to\ 2x^2-4x=0\ \to\ x^2-2x=0}\\ \mathrm{x(x-2)=0\ \ \| \ \ x_1=0\ \ \|\ \ x_2=2\ \ \| \ \ \mathbf{f(x)\cap g(x)=\{0,2\}}}$

$\mathrm{*\ Para\ y=0:}\\ \mathrm{y=2-x\ \to\ 0=2-x\ \to\ x=2\ \to\ \mathbf{(2,0)}}\\\\ \mathrm{*\ Para\ y=2:}\\ \mathrm{y=2-x\ \to\ 2=2-x\ \to\ x=0\ \to\ \mathbf{(0,2)}}\\\\ \mathbf{Pontos\ de\ intersec\c{c}\~ao\ \to\ \boxed{\mathbf{A(0,2)\ e\ B(2,0)}}}\\\\ \textbf{O comprimento da corda ser\'a dado\ pelo}\\ \textbf{c\'alculo da dist\^ancia entre os dois pontos:}\\\\ \mathrm{d_{AB}=\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}=}\\ \mathrm{=\sqrt{(2-0)^2+(0-2)^2}=\sqrt{8}\ \to\ \boxed{\mathbf{d_{AB}=2\sqrt{2}}}}$

$\textbf{Coordenadas do ponto m\'edio da corda:}\\\\ \mathrm{x_M=\dfrac{x_A+x_B}{2}=\dfrac{0+2}{2}=1}\\\\ \mathrm{y_M=\dfrac{y_A+y_B}{2}=\dfrac{2+0}{2}=1}\\\\ \mathbf{\boxed{\mathbf{(x_M,y_M)=(1,1)}}}$