What is cross section plan?

A satellite dish has a parabolic cross section with a focus that is 3 feet from the vertex. The cross section?

• Part A: Find the coordinates of the focus and the equation of the directrix. Explain your answers. Part B: Write an equation for the cross section of the satellite dish. Explain your answers. Part C: If the satellite dish is 4 feet deep, find the diameter of the satellite dish at its opening. Part D: If the opening of the satellite dish has a circumference of , how deep is the dish?

• Answer:

  lets take a xy plane , in which the origin is the vertex ; so vertex is O(0; 0) then the focus is F(0; 3) (in feet) then the directrix passes through point A(0; -3) and is parallel to the x-axis; then : (D) : y= -3 we have for vertex : (h;k) = (0; 0) and p=OF=3 then if we have : (x - h)² = 4p(y - k) then : (x-0)²= 4*3(y-0) x²=12y y=x²/12 if we take : depth=OB=4 ft then B(0; 4) so the perpendicular to y-axis in B is : (D') : y=4 and (D') crosses the parabola in : 4=x²/12 ⇔ x²=48 ⇔ x=±√48=±4√3 then the diameter equals : 4√3 * 2 = 8√3 ≈ 13.86 ft for the last part, sorry, the circumference is missing