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Two points O(0,0) and A(3,sqrt(3)) with ...

Two points `O(0,0)` and `A(3,sqrt(3))` with another point `P` form an equilateral triangle. Find the coordinates of `Pdot`

Text Solution

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Let the coordinates of P be (x,y).
`therefore OA=OP=AP`
or `OA^2=OP^2=AP^2`
`therefore 12=x^2+y^2=(x-3)^2+(y-sqrt3)^2`
`rArr3x+sqrt3y=6`
`rArrx=2-(y)/(sqrt3)` .....(1)
`therefore (2-(y)/(sqrt3))+y^2=12`
`rArry^2-sqrt3y-6=0`
`rArr(y-2sqrt3)(y+sqrt3)=0`
Form (1) , when `y=2sqrt3,x=0` and when `y=-sqrt3,x=3`
Hence, the coordinates of P are :
`(0,2sqrt3)` or `(3,-sqrt3)`
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