 Hello and welcome to the session. Let us discuss the following question. It says an equilateral triangle is inscribed in the parabola y square is equal to 4ax where one vertex is at the vertex of the parabola. Find the length of the side of the triangle. So we are given an equilateral triangle inscribed in the parabola. Now since OPQ is an equilateral triangle angle each angle of this triangle is 60 degrees. So angle P is equal to angle Q is equal to angle O is equal to 60 degrees. Now we see that the line joining PQ is perpendicular to the x axis. So PQ is perpendicular to O x. So PBO is the right triangle. Now let the coordinates of the point be x1, y1. So the point P x1, y1 lies on the parabola. Therefore this point satisfies the equation of the parabola. Therefore y1 square is equal to 4ax1 and this implies x1 is equal to y1 square upon 4a. Now considering the triangle P OB or OPB it is a right triangle angle, right angle at B and angle P is 60 degrees. So tan 60 degrees is equal to perpendicular upon base that is OB upon PB. Now OB is x1, PB is y1. So tan 60 degrees is equal to x1 upon y1. Now we know that tan 60 degrees is root 3. So root 3 is equal to x1 upon y1 and this implies x1 is equal to root 3, y1. Now we know that x1 is y1 square upon 4a. In A we have y1 square upon 4a is equal to root 3, y1 and this implies y1 is equal to 4 root 3a which is equal to PB. Now similarly considering the triangle OBQ, angle Q is 60 degrees and this is a right triangle angle, right angle at B. So you will find that BQ is equal to 4 root 3a. Let's call this as 1 and this as 2. So PQ is equal to PB plus BQ. Now PB is 4 root 3a, BQ is also 4 root 3a. So the sum is 8 root 3a. Now since this is an equilateral triangle all the sides equal. So PQ is equal to OP is equal to OQ is equal to 8 root 3a. Hence length of each side of triangle is 8 root 3a. And this completes the question and the session. Bye for now. Take care. Have a good day.