Question

If \((2,0)\) is the vertex and \(y\)-axis is the directrix of a parabola, then its focus is

• A. \((2,0)\)
• B. \((-2,0)\)
• C. \((4,0)\)
• D. \((-4,0)\)

Hint:

The vertex of a parabola is midway between its focus and directrix. If the directrix is \(x=0\) and vertex is \((2,0)\), then focus is \((4,0)\).

Answer 1

The Correct Option is C

We are given that the vertex of the parabola is \[ (2,0). \] Also, the directrix is the \(y\)-axis. The equation of the \(y\)-axis is \[ x=0. \] For a parabola, the vertex lies exactly midway between the focus and the directrix. The vertex is at \[ x=2. \] The directrix is \[ x=0. \] So the horizontal distance from the vertex to the directrix is \[ 2-0=2. \] Therefore, the focus must be at the same distance on the opposite side of the vertex. Since the directrix is to the left of the vertex, the focus will be to the right of the vertex. Thus, the \(x\)-coordinate of the focus is \[ 2+2=4. \] The \(y\)-coordinate remains \(0\), because the axis of the parabola is horizontal. Hence, the focus is \[ (4,0). \]