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MATH / In Cartesian space, three spheres centered at (−2,5,4), (2,1,4), and (4,7,5) are all tangent to the xy-plane. The xy-plane is one of two planes tangent to all three spheres; the se…

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problem

In Cartesian space, three spheres centered at (−2,5,4),(-2,5,4), (2,1,4),(2,1,4), and (4,7,5)(4,7,5) are all tangent to the xyxy-plane. The xyxy-plane is one of two planes tangent to all three spheres; the second plane can be written as the equation ax+bx+cz=dax + bx + cz = d for some real numbers a,a, b,b, c,c, and d.d. Find ca.\frac{c}{a}.
Plain-text mathematical notation (without MathML)
In Cartesian space, three spheres centered at (−2,5,4), (2,1,4), and (4,7,5) are all tangent to the xy-plane.  The xy-plane is one of two planes tangent to all three spheres; the second plane can be written as the equation ax+bx+cz=d for some real numbers a, b, c, and d.  Find (c)/(a).
Original LaTeX notation
In Cartesian space, three spheres centered at $(-2,5,4),$ $(2,1,4),$ and $(4,7,5)$ are all tangent to the $xy$-plane.  The $xy$-plane is one of two planes tangent to all three spheres; the second plane can be written as the equation $ax + bx + cz = d$ for some real numbers $a,$ $b,$ $c,$ and $d.$  Find $\frac{c}{a}.$

level

Level 5

type

Precalculus

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Source and history

task_id
1e10cc26-ac78-5595-b66f-eda175fff3b0
task_key
precalculus--test--1e10cc26-ac78-5595-b66f-eda175fff3b0
task_revision_id
1
config / split
precalculus / test

Official source

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