Question

The distance between the points $(t_1^3,\sqrt{2}{t}_1)$ and $(-t_2^3,-\sqrt{2}{t}_2),$ if $t_1$ and $t_2$ are the roots of $x^2-2\sqrt{2}x+1=0$ is

• A. $3$
• B. $6$ $\sqrt{6}$
• C. $5$ $\sqrt{2}$
• D. $10$

Answer 1

The correct option is B $6$ $\sqrt{6}$

Given equation

$x^2-2\sqrt{2}x+1=0$

$t_1+t_2=-\frac{-2\sqrt{2}}{1}=2\sqrt{2}{t}_1{t}_2=1$

distance between the points

$\sqrt{{({t_1}^3+{t_2}^3)}^2{+(\sqrt{2}{t}_1+\sqrt{2}{t}_2)}^2}=\sqrt{{\left((t_1+t_2)({t_1}^2-t_1{t}_2+{t_2}^2)\right)}^2+2{\left(t_1{+t}_2\right)}^2}=\sqrt{{\left((t_1+t_2)({(t_1+t_2)}^2-3t_1{t}_2)\right)}^2+2{\left(t_1{+t}_2\right)}^2}=\sqrt{{\left(\left(2\sqrt{2}\right)({\left(2\sqrt{2}\right)}^2-3)\right)}^2+2{\left(2\sqrt{2}\right)}^2}=\sqrt{200+16}=\sqrt{216}=6\sqrt{6}$

so option B is correct