conic sectionsgeometrymultivariable-calculus
I have a problem I am trying to solve, but I have no idea how to solve it.
If I have 3 spheres, $A(1, 2, 0), B(4, 5, 0), \text{ and } C(1, 3, 2)$ of radius 1, how would I go about finding the equation of the form $ax + by +cz = d$ of the plane which touches each of the spheres on the same side..?
I have been racking my brain all day, but I can't seem to figure it out.
If someone wouldnt mind lending a hand, It would be much appreciated..
Thanks
Corey
Notice, the dimensions of the box $a$, $b$ & $c$ are changing w.r.t. time $t$ hence, we have
substitute all the corresponding values, the rate of change of diagonal is $$\color{red}{\frac{dD}{dt}}=\frac{1}{\sqrt{1^2+2^2+3^2}}\left(1\cdot 1+2\cdot 1+3\cdot (-3)\right)=\color{red}{\frac{-6}{\sqrt {14}}\ m/s}$$
Join the center of the bigger circle (radius assumed to be $r$) to the mid-points of the square. It’s easy to see that $ABCD$ is a square as well. Now, join the center of the big circle to the center of one of the smaller circles ($P$). Then $BP=r+1$. Further, if we draw a vertical line through $P$, it intersects $AB$ at a point distant $r-1$ from $B$. Lastly, the perpendicular distance from $E$ to the bottom side of the square is equal to $AD=r$. Take away three radii to obtain $EP=r-3$. Using Pythagoras’ Theorem, $$(r-1)^2 +(r-3)^2 =(r+1)^2 \\ r^2-10r+9=0 \implies r=9,1$$, but clearly $r\ne 1$, and so the side of the square is $2r=18$.
Best Answer
If they all have the same radius, the problem can be easily solved. The plane needs to be parallel to the plane that contains the three centers; This yields the parameters $a, b, c$. Choosing the right distance to the plane that contains the centers gives you the two possible values for $d$.