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  "body": "\n## Definition\n\nLet $K$ be a right circular [[cone|7TPBQPPN97]] in $\\R^3$ with vertex $V$.\nLet $\\pi$ be a plane not through $V$ that meets $K$ in a [[conic section|00HJCMC2CD]] $C$.\nA *Dandelin sphere* of $K$ and $\\pi$ is a [[sphere|P9CM4GQ2T0]] $\\Sigma$ such that:\n\n1. $\\Sigma$ is tangent to $K$ along a circle, that is, $\\Sigma$ is inscribed in $K$;\n2. $\\Sigma$ is tangent to the plane $\\pi$ at one point.\n\n## Remarks\n\nWhen $C$ is an [[ellipse|0VKCCEXBJ1]] there are exactly two Dandelin spheres, one on each side of $\\pi$.\nFor a hyperbola there are two, one in each nappe, and for a parabola there is one.\nCheck of the focal property for an ellipse: let $\\Sigma_1, \\Sigma_2$ touch $\\pi$ at $F_1, F_2$ and touch $K$ along circles $c_1, c_2$.\nFor $P \\in C$, the generator line $VP$ meets $c_1$ and $c_2$ at points $Q_1, Q_2$.\nTwo tangent segments from $P$ to the same sphere have the same length, so $PF_1 = PQ_1$ and $PF_2 = PQ_2$.\nHence $PF_1 + PF_2 = Q_1 Q_2$, which is the distance between the parallel circles $c_1$ and $c_2$ measured along a generator, the same for all $P$; so $C$ is an ellipse with foci $F_1, F_2$.\nWhen $C$ is not a circle, the plane of each circle $c_i$ meets $\\pi$ in a line, which is the directrix belonging to the focus $F_i$.\nNon-example: take a small sphere inscribed in $K$ near the vertex, on the vertex side of $\\pi$ and far from it.\nIt is tangent to $K$ along a circle but does not touch $\\pi$, so clause 2 fails.\n",
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