An equilateral triangle is inscribed in the parabola y 2 = 4 ax , where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
It is given that equilateral triangle is inscribed in the parabola y 2 = 4 a x and one vertex is at the parabola.
Let, OAB be the equilateral triangle inscribed in the parabola.
Let, OC = k .
Now, the equation of parabola is,
y 2 = 4 a k y = ± 2 a k
Since ( k , ± 2 k ) satisfies the equation of the parabola,
y 2 = 4 a k y = ± 2 a k
The respective coordinates of A and B are ( k , 2 k ) and ( k , − 2 k ) .
AB = CA + CB = 2 a k + 2 a k = 4 a k
Since OAB is an equilateral triangle,
( OA ) 2 = ( AB ) 2 k 2 + ( 2 a k ) 2 = ( 4 a k ) 2 k 2 + 4 a k = 16 a k k = 12 a
Substitute the value of k in 4 a k .
4 a k = 4 a × 12 a = 4 12 a 2 = 8 3 a
Thus, the side of equilateral triangle is 8 3 a .
It is given that equilateral triangle is inscribed in the parabola
Let, OAB be the equilateral triangle inscribed in the parabola.
Let,
Now, the equation of parabola is,
Since
The respective coordinates of
Since
Substitute the value of
Thus, the side of equilateral triangle is
