Question

If points $P(x,y),O(0,0)$ and $A(3,\sqrt{3})$ forms an equilateral triangle then Find the coordinates of $P$?

Answer 1

Two vertices of an equilateral triangle are (0, 0) and (3, √3).
Let the third vertex of the equilaterla triangle be (x, y)
Distance between (0, 0) and (x, y) = Distance between (0, 0) and (3, √3) = Distance between (x, y) and (3, √3)
√(x² + y²) = √(3² + 3) = √[(x - 3)² + (y - √3)²]
x² + y² = 12
x² + 9 - 6x + y² + 3 - 2√3y = 12
24 - 6x - 2√3y = 12
- 6x - 2√3y = - 12
3x + √3y = 6
x = (6 - √3y) / 3
⇒ [(6 - √3y)/3]² + y² = 12
⇒ (36 + 3y² - 12√3y) / 9 + y² = 12
⇒ 36 + 3y² - 12√3y + 9y² = 108
⇒ - 12√3y + 12y² - 72 = 0
⇒ -√3y + y² - 6 = 0
⇒ (y - 2√3)(y + √3) = 0
⇒ y = 2√3 or - √3
If y = 2√3, x = (6 - 6) / 3 = 0
If y = -√3, x = (6 + 3) / 3 = 3
So, the third vertex of the equilateral triangle = (0, 2√3) or (3, -√3).