Question

Find the points having x-coordinate of 9 whose distance from the point (3,-2) is 10.

Answer 1

$\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{3}~,~\stackrel{y_1}{-2})\qquad (\stackrel{x_2}{9}~,~\stackrel{y_2}{y})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ \stackrel{d}{10}=\sqrt{[9-3]^2+[y-(-2)^2]}\implies 10=\sqrt{(9-3)^2+(y+2)^2} \\\\\\ 10=\sqrt{6^2+\stackrel{FOIL}{(y^2+4y+4)}}\implies 10^2=36+(y^2+4y+4)$

$\bf 100=y^2+4y+40\implies 0=y^2+4y-60 \\\\\\ 0=(y+10)(y-6)\implies y= \begin{cases} -10\\ 6 \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill (9,-10)\qquad (9,6)~\hfill$