Question

For what values of a is the distance between P(a,3) and Q(5,2a) greater than √26

Answer 1

$P(a,3)\ \ \ Q(5,2a)\\\\ PQ=\sqrt{(x_Q-x_P)^2+(y_Q-y_P)^2}\\\\ PQ>\sqrt{26}\\\\\sqrt{(5-a)^2+(2a-3)^2}>\sqrt{26}\\\\\sqrt{25-10a+a^2+4a^2-12a+9}>\sqrt{26}\\\\\sqrt{5a^2-22a+34}>\sqrt{26}\ \ |^{2}\\\\ 5a^2-22a+34=26\ \ \ |Subtract\ 26\\\\ 5a^2-22a+8=0\ \ \ |Factoring\\\\(a-4)(5a-2)=0\\\\ a=4\ \ or\ \ 5a=2\\ a=4\ \ or\ \ a=\frac{2}{5}$