In which of the following cases, a unique parabola will be obtained ?

Options

A. (a) Focus and equation of tangent at vertex are given.
B. (b) Focus and vertex are given
C. (c) Equation of directrix and vertex are given.
D. (d) Equation of directrix and equation of tangent at vertex are given.

Answer

A, B, C

Explanation

1,2,3
(1) Focus and equation of tangents at vertex are given.
Directrix is parallel to tangent at vertex.
Also, distance between directrix and tangent at vertex.
So, equation of directrix and tangents at vertex.
Therefore, parabola is unique.
(2) Focus and vertex are given.
Distance between focus and vertex is 'a' .
Also distance between vertex and directrix is 'a'.
Directrix is perpendicular to the axis which passes through focus and vertex.
So, equation of directrix can be determined.
Therefore, parabola is unique.
(3) Equation of directrix and vertex can be determined.
Therefore, parabola is unique.
(4) Equation of directrix and equation of tangent at vertex are given.
Here, only distance between these two lines can be found, four times of which is latus rectum.
Therefore, parabola is not unique.