Question

A distância entre os pontos A(-2, y) e B(6, 7) é 10. O valor de y é:  ​

Answer 1

Resposta:

y+ = 13 ou y- = 1

Explicação passo a passo:

Distância entre dois pontos:

$10^{2} = (x_{b} - x_{a})^{2} + (y_{b} - y_{a})^{2} \\\\100 = (6_{} - (-2)_{})^{2} + (7_{} - y_{})^{2} \\\\100 = (6_{} +2)_{}^{2} + (7_{} - y_{})^{2} \\\\\ 100 = (8)^{2} + (7_{} - y_{})^{2} \\\\100 = 64 + (7_{} - y_{})^{2} \\\\100 = y^{2} - 14y + 113 \\\\0= y^{2} - 14y + 113 - 100 \\\\y^{2} - 14y + 13 = 0$

Utilizando a fórmula de Bhaskara para o cálculo das raízes:

$y =\frac{ -b +- \sqrt{b^{2} -4ac} }{2} \\\\y =\frac{ -(-14) +- \sqrt{-14^{2} -4.1.13} }{2} \\\\y =\frac{ 14 +- \sqrt{-14^{2} -52} }{2} \\\\y =\frac{ 14 +- \sqrt{196 -52} }{2} \\\\y =\frac{ 14 +- \sqrt{144} }{2} \\\\y =\frac{ 14 +-12}{2} \\\\\\y+ =\frac{ 14 +12}{2} \\\\y+ =\frac{ 26}{2} \\\\y+ = 13\\\\\\y- =\frac{ 14 -12}{2} \\\\y- =\frac{ 2}{2} \\\\y- = 1$