Find the equation of parabola (i) having focus at (0,-3) its directrix is y = 3. (ii) having end points of latus rectum (5,10) and (5,-10) and which opens towards right. (iii) having vertex at origin and focus at (0,2)

Explanation

(i) Here, focus S(0,-3) and directrix y=3 are at same distance from origin is vertex of the parabola and parabola opens downwards.
So, we consider equation `x^(2)=-4ay`.
Here focus is `(0,-3)-=(0,-a)`
`:." "a=3`
So, equation of parabola is `y^(2)=-12x`.
(ii) End points of latus are A (5,10) and B (5,-10).
`:." "S-=(5,0)`
Also, parabola opens towards right.
So, we consider equation `y^(2)=4ax`.
`:.` Vertex is (0,0) and a=5.
So, equation of parabola is `y^(2)=20x`.
(iii) Vertex is (0,0) focus S(0,2).
So, directrix is y=-2.
So, we consider equation `x^(2)=4ay`, where a=2.
`:.` Equation of parabola is `x^(2)=8y`.