Tough question.
The first part of the solution:Let the origin be O.
The area of the triangle ABC is 1/8.
The area of the triangle ABC = the area of the triangle AOB*2
The area of the triangle AOB = (distance between Origin and B * distance between Origin and A)/2
The area of the triangle ABC = ((distance between Origin and B * distance between Origin and A)/2)*2 =>
The area of the triangle ABC = distance between Origin and B * distance between Origin and A = 1/8
The second part of the solution:Point A is the point when x=0
If x=0, then y=k-0 => y=k
So, we now know that the distance between the origin and the point A is
k.
Points B and C are points when y=0
0=k-x^2 => x^2 = k => x = +- sqrt(k)
So, we now know that the distance between the origin and the point B is
sqrt(k)The third part of the solution:k*sqrt(k) = 1/8
k^3 = 1/64
k = 1/sqrt(8)
k = 0.35
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