Question

Encontre o valor de t cuja distância entre os pontos (2, 3) e (t, 2t) é a menor possível.​

Answer 1

Se queremos a menor distância se trata de um mínimo, logo usaremos derivada em distância entre dois pontos :

$\displaystyle \sf D =\sqrt{(t-2)^2+(2t-3)^2} \\\\ D=\sqrt{t^2-4t+4+4t^2-12t+9 } \\\\ D =\sqrt{5t^2- 16t+13} \\\\ \text{menor dist{\^a}ncia : derivada igual a 0 } : \\\\ D_{\text{m{\'i}nimo}} = [\sqrt{5t^2- 16t+13}]' = 0\\\\\\ ((5t^2- 16t+13)^{\frac{1}{2}})'=0 \\\\\\ \frac{1}{2}\cdot (5t^2- 16t+13)^{\left(\frac{1}{2}-1\right)}\cdot (5t^2- 16t+13)'=0$

$\displaystyle \sf \\\\\\ \frac{10t-16}{2\cdot \sqrt{5t^2- 16t+13}} = 0\\\\\\\ \frac{5t-8}{\sqrt{5t^2- 16t+13}} \\\\\\ \text{Denominador }: \\\\ \sqrt{5t^2- 16t+13}\neq 0 \\\\ \text{numerador} : \\\\ 5t-8 = 0 \\\\ \huge\boxed{\sf t = \frac{8}{5} \ }\checkmark$