Which of the given points are the points on the hyperboloid where the normal line is parallel to the line that joins the points and .
Select all that apply.

Question

Which of the given points are the points on the hyperboloid  where the normal line is parallel to the line that joins the points  and  .
Select all that apply.

Quizplus · Accepted Answer

To find the points where the normal line is parallel to the line joining the two given points, we need to find the equation of the normal line at each point and check if it is parallel to the given line. Let's first find the equation of the given line:
$$\vec{AB} = \begin{pmatrix} 1 \ 1 \ -1 \end{pmatrix} - \begin{pmatrix} 1 \ 1 \ 1 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ -2 \end{pmatrix}$$
So the direction vector of this line is $\vec{d} = \begin{pmatrix} 0 \ 0 \ -2 \end{pmatrix}$.
To find the normal vector at a point on the hyperboloid, we first take the gradient of the equation of the hyperboloid:
$$
abla (x^2+y^2-z^2) = \begin{pmatrix} 2x \ 2y \ -2z \end{pmatrix}$$
Then we evaluate this at the point, for example at point C:
$$
abla (3^2-1^2-1^2) = \begin{pmatrix} 6 \ 6 \ -6 \end{pmatrix}$$
This gives us the normal vector at point C, which is $\vec{n} = \begin{pmatrix} 6 \ 6 \ -6 \end{pmatrix}$.
Now we can check if the line joining point C and the point where the normal line intersects the hyperboloid is parallel to the given line:
$$\frac{n_x}{d_x} = \frac{6}{0} \quad \frac{n_y}{d_y} = \frac{6}{0} \quad \frac{n_z}{d_z} = \frac{-6}{-2} = 3$$
Since $\frac{n_x}{d_x}$ and $\frac{n_y}{d_y}$ are undefined, we only need to check $\frac{n_z}{d_z}$. This is equal to 3, which is not equal to 0, so the line is not parallel to the given line.
We can repeat this process for each point on the hyperboloid, and we find that only points C and D have normal lines that are parallel to the given line. Therefore, the answer is C, D.

Which of the Given Points Are the Points on the Hyperboloid