Resposta:
[tex]\textsf{Leia abaixo}[/tex]
Explicação passo a passo:
[tex]\mathsf{A(4,2)\:\:\:\:\:B(1,5)\:\:\:\:\:C(-3,4)\:\:\:\:\:P(x,y)}[/tex]
[tex]\mathsf{d_{AP} = d_{BP} = d_{CP}}[/tex]
[tex]\begin{cases}\mathsf{(x_A - x_P)^2 + (y_A - y_P)^2 = (x_B - x_P)^2 + (y_B - y_P)^2}\\\mathsf{(x_B - x_P)^2 + (y_B - y_P)^2 = (x_C - x_P)^2 + (y_C - y_P)^2}\end{cases}[/tex]
[tex]\begin{cases}\mathsf{(4 - x)^2 + (2 - y)^2 = (1 - x)^2 + (5 - y)^2}\\\mathsf{(1 - x)^2 + (5 - y)^2 = (-3 - x)^2 + (4 - y)^2}\end{cases}[/tex]
[tex]\mathsf{16 - 8x + x^2 + 4 - 4y + y^2 = 1 - 2x + x^2 + 25 - 10y + y^2 }[/tex]
[tex]\mathsf{- 8x - 4y + 20 = - 2x - 10y + 26}[/tex]
[tex]\mathsf{-x + y = 1}[/tex]
[tex]\mathsf{1 - 2x + x^2 + 25 - 10y + y^2 = 9 + 6x + x^2 + 16 - 8y + y^2}[/tex]
[tex]\mathsf{-2x - 10y + 26 = 6x - 8y + 25}[/tex]
[tex]\mathsf{-8x - 2y = -1}[/tex]
[tex]\begin{cases}\mathsf{x - y = -1}\\\mathsf{8x + 2y = 1}\end{cases}[/tex]
[tex]\mathsf{10x = -1}[/tex]
[tex]\mathsf{x = -\dfrac{1}{10}}[/tex]
[tex]\mathsf{8\left(-\dfrac{1}{10}\right) + 2y = 1}[/tex]
[tex]\mathsf{-\dfrac{4}{5} + 2y = 1}[/tex]
[tex]\mathsf{-4 + 10y = 5}[/tex]
[tex]\mathsf{10y = 9}[/tex]
[tex]\mathsf{y = \dfrac{9}{10}}[/tex]
[tex]\mathsf{x + y = -\dfrac{1}{10} + \dfrac{9}{10}}[/tex]
[tex]\mathsf{x + y = \dfrac{8 \div 2}{10 \div 2}}[/tex]
[tex]\boxed{\boxed{\mathsf{x + y = \dfrac{4}{5}}}}\leftarrow\textsf{letra C}[/tex]
