This is a question I have for an assignment and I have no idea how to get started. Help wanted! :)
Let S be the sphere centered at $(3, 0, 4)$ and of radius $2\sqrt 3$. Determine the pointnormal equations of the planes tangent to the sphere $S$ and orthogonal to the line $L : (x, y, z) = (4, 11, 7) + t(1, −1, 1)$. (ON-system)
Hint:
Take a line $R$ parallel to $L$ and passing through the center of the sphere.
Take the intersections of this line with the sphere (two points)
The planes tangent to the sphere at these two points are orthogonal to the line $R$ and also to the line $L$.
Added:
The line $R$ has equation: $(x,y,z)^T=(3,0,4)^T+t(1,-1,1)^T \qquad (1)$
The sphere has equation: $(x-3)^2+y^2+(z-4)^2=12 \qquad (2)$
To find the points of intersection substitute $(1)$ in $(2)$, this gives: $$ [(3+t)-3]^2+(-t)^2+[(4+t)-4]^2=12 \iff t^2 = 4 \iff t=\pm 2 $$ The points of intersection of $R$ with the sphere are $A=(5,-2,6)^T$ and $B=(1,2,2)^T$
Since the vector direction of $R$ is $\vec r=(1,-1,1)^T$ the plane passing in $A$ and orthogonal to $R$ has equation:
$$ (\vec x-\vec A) \cdot \vec r=0 \iff (x-5)-(y+2)+(z-6)=0 $$
so the equation of this plane is $x-y+z=13$. And in the same manner you can find the other plane corresponding to $t=-2$.