Question

Determine k de modo que a distância entre A(3,7) e B(k,12) seja igual a 13.

Answer 1

$(d_{AB})^{2}=(x_{A}-x_{B})^{2}+(y_{A}-y_{B})^{2}\\13^{2}=(3-k)^{2}+(7-12)^{2}\\169=(3^{2}-2\cdot3\cdot k+k^{2})+(-5)^{2}\\169=9-6k+k^{2}+25\\k^{2}-6k+34=169\\k^{2}-6k+34-169=0\\k^{2}-6k-135=0\\\\\Delta=b^{2}-4ac\\\Delta=(-6)^{2}-4\cdot1\cdot(-135)\\\Delta=36+540\\\Delta=576\\\\k=\dfrac{-b\pm\sqrt{\Delta}}{2a}~~~\therefore~~~\dfrac{-(-6)\pm\sqrt{576}}{2\cdot1}~~~\therefore~~~\dfrac{6\pm24}{2}~~~\therefore~~~3\pm12$

Temos 2 valores possiveis pra k:

$k=3+12=15\\k=3-12=-9$