I have a sphere with radius $d$ centered at the origin. The $z$-axis is vertical. I take any point $P$ on that sphere, $P(d \cos\phi \sin\theta,\ d \sin\phi \sin\theta,\ d \cos\theta)$. I take a plane that is tangent to the sphere, and its point of tangency is at point $P$. Therefore, the equation of the plane would be $x \cos\phi \sin\theta + y \sin\phi \sin\theta + z \cos\theta = d$. Point $P_1$ lies on that same plane. Point $P$ is $r$ units away from point $P_1$. Point $P$ has the same $z$-coordinate as point $P_1$, and I am trying to find the $x$-coordinate and $y$-coordinate of point $P_1$, while still preserving the variables. Is this enough information to find these coordinates? If not, what other information do I need to provide?
Thank you in advance.
Best Answer
In the plane tangent at point $P$, point $P_1$ can be on a circle with radius $r$, centered on point $P$. You have $z=\cos\theta$ as a plane. The intersection of two planes is usually a line that belongs to both planes. Since you know that $P$ is on the line, you get a line passing through the center of the circle. Then the intersection will consist of two points, diametrically opposed. But this is the case when $\theta$ is $0$ or $\pi$. Then the tangent plane is at constant $z=\pm1$, so the solution is any point with coordinates $(r\cos\alpha,r\sin\alpha, \pm1$) is fine.