Let points A 1 , A 2 and A 3 lie on the parabola y 2 = 8 x . If △ A 1 A 2 A 3 is an equilateral triangle and normals at points A 1 , A 2 and A 3 on this parabola meet at the point h , 0 , then the value of h is

Question

Let points A 1 , A 2 and A 3 lie on the parabola y 2 = 8 x . If △ A 1 A 2 A 3 is an equilateral triangle and normals at points A 1 , A 2 and A 3 on this parabola meet at the point h , 0 , then the value of h is, 24, 26, 38, 28

MARKS App · Accepted Answer

One of the normal from h , 0 on the parabola y 2 = 8 x is y = 0 hence, assume A 3 = 0 , 0 Let, A 1 = 2 t 1 2 , 4 t 1 and A 2 = 2 t 1 2 , - 4 t 1 Then, the slope of O A 1 is 4 t 1 2 t 1 2 = 2 t 1 ⇒ tan ⁡ 30 o = 2 t 1 ⇒ t 1 = 2 3 The equation of the normal at A 1 is y = - t 1 x + 4 t 1 + 2 t 1 3 ⇒ y = - 2 3 x + 8 3 + 48 3 Putting h , 0 in the above equation, we get, 0 = - 2 3 h + 56 3 ⇒ h = 28 .

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