∵OA、OB的方程分别为y=根号3x和y＝－根号3x（x>=0)
又：tanπ/3=根号3
∴OA,OP与x轴的夹角分别为π/3,-π/3
连接OP
设OP = r,OP与x轴夹角为α,α∈【-π/3,π/3】
∠MOP = π/3-α
∠MOP = -π/3+α
OM = OP cos(π/3-α)
ON = OP cos(-π/3+α)
SOMPN = S△OMP + S△OPN
= | 1/2 *OM*OP*sin(π/3-α) | + | 1/2 *ON*OP*sin(-π/3+α) | 【| | 是绝对值】
= | 1/2 *OP cos(π/3-α)*OP*sin(π/3-α) | + | 1/2 *OP cos(-π/3+α)*OP*sin(-π/3+α) |
= | 1/4 *OP^2*sin(2π/3-2α) | + | 1/4 *OP^2 sin(-2π/3+2α) |
= 1/4 r^2 { | sin(2π/3-2α) | + | sin(-2π/3+2α) | }
= 1/4 r^2 { |- sin[-(2π/3-2α) | + | sin(-2π/3+2α) | }
= 1/4 r^2 { | sin(2α-2π/3) | + | sin(2α-2π/3) | }
= 1/4 r^2 * 2 | sin(2α-2π/3) |
= 1/2r^2 * | sin(2α-2π/3) |
= 1/2r^2 * | sin(2α)cos(2π/3)-cos(2α)sin(2π/3) |
= 1/2r^2 * | sin(2α)(-1/2)-cos(2α)*根号3/2 |
= 1/4 r^2 | sin2α+根号3cos2α |
= 1/4 | 2rsinα*rcosα+根号3*(rcosα)^2-根号3*(rsinα)^2 | = 根号3
将rcosα=x,rsinα=y代入上式得：
1/4 |2xy+根号3 x^2 - 根号3 y^2 | = 根号3
两边同乘以4根号3：
| 3x^2 + 2根号3 xy -3y^2| = 12