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Misc 8
An equilateral triangle is inscribed in the parabola y2 = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
Let length of equilateral triangle be s
Hence OA = OB = AB = s
Here, OC AB
So, OCA = OCB = 90
And AC = BC
So, AC = BC = 2
AC = BC = 2
We find coordinates of point B
Now,
in right triangle OBC
Using Pythagoras theorem
(Hypotenuse)2 = (Height)2 + (Base)2
(OB)2 = (OC)2 + (BC)2
s2 = (OC)2 + 2 2
s2 = (OC)2 + 2 4
s2 2 4 = (OC)2
4 2 2 4 = (OC)2
3 2 4 = (OC)2
(OC)2 = 3 2 4
OC = 3 2 4
OC = 3 2
Hence coordinate of point B is B( , )
Now, point B lies on parabola
So, it must satisfy its equation
Putting x = 3 2 , y = 2 in equation of parabola
y2 = 4ax
2 2 = 4a( 3 2 )
2 4 = 4a( 3 2 )
2 = 4 4a( 3 2 )
s = 8 a
Hence, side of equilateral triangle = 8 3 a

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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