From Problem 4(b) you know that $U \sim Chis(4).$ Independently, $Y = X_6^2 + X_7^2 \sim Chis(2).$ Thus in 3(c), $\frac{Y/2}{U/4} = \frac{2Y}{U} \sim F(2,4),$ Snedecor's F-distribution with numerator df 2 and denominator df 4. Your answer seems OK, except that you have put in an extra 4.
_Note:_ The F disstribution is useful for testing from data whether two normal distributions have equal variances. Also, in the analysis of variance. Fisher introduced the idea of this distribution, calling it the variance-ratio distribution; Snedecor put it into the form usually used today.
I assume you got Student's t-distributions in 3(a) and (b).