[tex]\displaystyle \sf x =4y-4 \ ; \\\\ x^2+4y^2=16 \ ;[/tex]

para determinar o comprimento da reta, faça distância entre dois pontos que a reta determina na elipse.
para achar os pontos, substitua a equação da reta na equação da elipse e resolva.
[tex]\displaystyle \sf x^2+4y^2=16 \\\\ (4y-4)^2+4y^2=16 \\\\ 16y^2-32y+16 +4y^2=16 \\\\ 20y^2-32y =0 \\\\ y(20y-32) = 0 \\\\ y = 0 \\\\ 20y-32=0 \\\\ y=\frac{32}{20} \to y = \frac{8}{5} \\\\\\ \text{substituindo na equa\c c\~ao da reta} :\\\\ x = 4.0-4 \to x =-4\to P_1=(-4,0) \\\\ x=4.\frac{8}{5}-4 \to x=\frac{12}{5} \to P_2=\left(\frac{12}{5},\ \frac{8}{5}\right)[/tex]

Distância entre os pontos :

[tex]\displaystyle \sf \overline{P_1P_2} = \sqrt{\left(-4-\frac{12}{5}\right)^2+\left(0-\frac{8}{5}\right)^2} \\\\\\ \overline{P_1P_2} = \sqrt{\left(\frac{-20-12}{5}\right)^2+\left(-\frac{8}{5}\right)^2} \\\\\\ \overline{P_1P_2} = \sqrt{\frac{32^2}{25}+\frac{64}{25}} =\sqrt{\frac{1024+64}{25}} \\\\\\\ \overline{P_1P_2} =\frac{\sqrt{1088}}{5}=\frac{\sqrt{2^6\cdot 17}}{5} \\\\\\ \overline{P_1P_2} =\frac{2^3\sqrt{17}}{5}[/tex]

portanto o comprimento da corda é:

[tex]\large\boxed{\sf \ \overline{P_1P_2} =\frac{8\sqrt{17}}{5} \ }\checkmark[/tex]