Question

Calcular o valor de y de modo que o ponto (1, y) seja equidistante dos pontos (1,0) e (0,2). ​

Answer 1

✅ Após ter devidamente calculado o valor de "y" que torna o ponto "M" equidistante de "A" e "B" é:

                     $\large\displaystyle\begin{gathered}y = \frac{5}{4} \end{gathered}$

Sejam os pontos:

                    $\large\begin{cases}A(1, 0)\\B(0, 2)\\M(1, y) \end{cases}$

Para que o ponto "M" seja equidistante de "A" e "B" é necessário que:

                                               $\large\displaystyle\text{ \begin{gathered}D_{\overline{AM}} = D_{\overline{MB}}\end{gathered} }$

   $\large\displaystyle\text{ \begin{gathered}\sqrt{(X_{M} - X_{A})^{2} + (Y_{M} - Y_{A})^{2}} = \sqrt{(X_{B} - X_{M})^{2} + (Y_{B} - Y_{M})^{2}} \end{gathered} }$

$\large\displaystyle\text{ \begin{gathered}(\sqrt{(X_{M} - X_{A})^{2} + (Y_{M} - Y_{A})^{2}})^{2} = (\sqrt{(X_{B} - X_{M})^{2} + (Y_{B} - Y_{M})^{2}})^{2} \end{gathered} }$

        $\large\displaystyle\begin{gathered}(X_{M} - X_{A})^{2} + (Y_{M} - Y_{A})^{2} = (X_{B} - X_{M})^{2} + (Y_{B} - Y_{M})^{2} \end{gathered}$

                       $\large\displaystyle\begin{gathered}(1 - 1)^{2} + (y - 0)^{2} = (0 - 1)^{2} + (2 - y)^{2} \end{gathered}$

                                            $\large\displaystyle\begin{gathered}0^{2} + y^{2} = (-1)^{2} + (2 - y)^{2} \end{gathered}$

                                                     $\large\displaystyle\begin{gathered}y^{2} = 1 + 4 - 4y + y^{2}\end{gathered}$

                                                     $\large\displaystyle\begin{gathered}y^{2} = 5 - 4y + y^{2} \end{gathered}$

                           $\large\displaystyle\begin{gathered}\!\diagup\!\!\!\!\!y^{2} - 5 + 4y - \!\diagup\!\!\1\!\!\!y^{2} = 0 \end{gathered}$

                                          $\large\displaystyle\begin{gathered}-5 + 4y = 0 \end{gathered}$

                                                    $\large\displaystyle\begin{gathered}4y = 5 \end{gathered}$

                                                      $\large\displaystyle\begin{gathered}y = \frac{5}{4} \end{gathered}$

✅ Portanto o valor de "y" que torna "M" equidistante de "A" e "B" é:

                                 $\large\displaystyle\begin{gathered}y = \frac{5}{4} \end{gathered}$

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