Question

Find the equation of the normal to the solid 2x
2 −y
2 +8z
2 = 11 at a point where it
intersects the line x−3 = z =
y+1
2
.

Answer 1

According to the following equation x-3=z=y+1 we can make a system of equations.$\begin{Bmatrix} x = t + 3 \\ y = t - 12\\ z = t \end{Bmatrix}$

And with substituting this values in first equation of solid we get:

2(t+3)²-(t-12)²+8(t)² = 11 \\transform a little

2t² + 18 + 12t - t² - 1 + 2t + 8t² - 11 = 0;

9t² + 14t + 6 = 0;

discriminant of this equation is D = 14*14 - 4*9*6 = -20 < 0 , So this means that there is no rational roots, then the solid does not intersect the line.