###### Answer: d. x^2 = 6y

Suppose a chord crosses a parabola perpendicular to its axis of symmetry. Let the length of the chord between the points where it intersects the parabola be c and the distance from the vertex of the parabola to the chord measured along the axis of symmetry be d. The focal length f of the parabola is given by ${\displaystyle f={\frac {c^{2}}{16d}}.}$ Proof Suppose a system of Cartesian coordinates is used such that the vertex of the parabola is at t…

Suppose a chord crosses a parabola perpendicular to its axis of symmetry. Let the length of the chord between the points where it intersects the parabola be c and the distance from the vertex of the parabola to the chord measured along the axis of symmetry be d. The focal length f of the parabola is given by ${\displaystyle f={\frac {c^{2}}{16d}}.}$ Proof Suppose a system of Cartesian coordinates is used such that the vertex of the parabola is at the origin and the axis of symmetry is the y axis. The parabola opens upward. It is shown elsewhere in this article that the equation of the parabola is 4fy = x where f is the focal length. At the positive x end of the chord x = c/2 and y = d. Since this point is on the parabola these coordinates must satisfy the equation above. Therefore by substitution ${\displaystyle 4fd=\left({\tfrac {c}{2}}\right)^{2}}$. From this ${\displaystyle f={\tfrac {c^{2}}{16d}}}$. The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram that surrounds it. One side of the parallelogram is the chord and the opposite side is a tangent to the parabola. The slope of the other parallel sides is irrelevant to the area. Often as here they are drawn parallel with the parabola’s axis of symmetry but this is arbitrary. A theorem equivalent to this one but different in details was derived by Archimedes in the 3rd century BCE. He used the areas of triangles rather than that of the parallelogram. See The Quadrature of the Parabola. If the chord has length b and is perpendicular to the parabola’s axis of symmetry and if the perpendicular distance from the parabola’s vertex to the chord is h the parallelogram is a rectangle with sides of b an…

The Quadrature of the Parabola (Greek: Τετραγωνισμὸς παραβολῆς) is a treatise on geometry written by Archimedes in the 3rd century BC. Written as a letter to his friend Dositheus the work presents 24 propositions regarding parabolas culminating in a proof that the area of a parabolic segment (the region enclosed by a parabola and a line) is 4/3 that of a certain …

Conic section – Wikipedia

Parabola – Wikipedia

Parabola – Wikipedia

Conic section – Wikipedia

Equivalently this is the grap…