Question

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

• A. Focus and equation of tangent at vertex are given.
• A. Equation of directrix and vertex are given.
• B. Focus and vertex are given.
• D. Equation of directrix and equation of tangent at vertex are given.

Answer 1

Answer: (A), (A), (B), (D)

(1), (2), (3), (4)

Answer 2

A parabola is uniquely defined by its focus and directrix. We analyze each case:

Case (1): Focus and equation of tangent at vertex are given. If the focus $F$ and the tangent at vertex $T_V$ are given, the axis of symmetry is the line through $F$ perpendicular to $T_V$. The vertex $V$ is the intersection of the axis and $T_V$. The distance $VF$ is the focal length $p$. The directrix is then determined as the line perpendicular to the axis, at a distance $p$ from $V$ on the opposite side of $F$. Thus, the focus and directrix are uniquely determined, resulting in a unique parabola.

Case (2): Focus and vertex are given. If the focus $F$ and vertex $V$ are given, the axis of symmetry is the line passing through $F$ and $V$. The distance $VF$ is the focal length $p$. The directrix is uniquely determined as the line perpendicular to the axis of symmetry, passing through a point $D'$ such that $V$ is the midpoint of the segment $FD'$. Thus, the focus and directrix are uniquely determined, resulting in a unique parabola.

Case (3): Equation of directrix and vertex are given. If the directrix $D$ and vertex $V$ are given, the axis of symmetry is the line through $V$ perpendicular to $D$. The distance from $V$ to $D$ is the focal length $p$. The focus $F$ is uniquely determined as the point on the axis of symmetry, at a distance $p$ from $V$, on the side opposite to the directrix $D$. Thus, the focus and directrix are uniquely determined, resulting in a unique parabola.

Case (4): Equation of directrix and equation of tangent at vertex are given. If the directrix $D$ and the tangent at vertex $T_V$ are given, for a parabola to exist, $D$ and $T_V$ must be parallel. Assuming they are parallel, the axis of symmetry is the line perpendicular to both $D$ and $T_V$. The vertex $V$ is the intersection of the axis and $T_V$. The distance from $V$ to $D$ is the focal length $p$. The focus $F$ is uniquely determined as the point on the axis of symmetry, at a distance $p$ from $V$, on the side opposite to the directrix $D$. Thus, the focus and directrix are uniquely determined, resulting in a unique parabola.

In all four cases, the focus and directrix of the parabola can be uniquely determined, which means a unique parabola is obtained.