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:

A quick way to find the focus is to think about the vertex as the "center". The directrix is on one side, and the focus is on the other, at an equal distance. Here, the vertex is at x=2, directrix at x=0 (2 units to the left). So, the focus must be 2 units to the right, at x=2+2=4.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the vertex and the directrix of a parabola and we need to find its focus.

Step 2: Key Formula or Approach:
The vertex of a parabola is the midpoint between its focus and its directrix.
The axis of the parabola is perpendicular to the directrix and passes through the vertex.
The distance from the vertex to the directrix is equal to the distance from the vertex to the focus. Let this distance be 'a'.

Step 3: Detailed Explanation:
The vertex is \(V(h, k) = (2, 0)\).
The directrix is the y-axis, which is the line \(x=0\).
Since the directrix is a vertical line (\(x=0\)), the axis of symmetry must be a horizontal line. Since the axis passes through the vertex (2,0), the axis is the x-axis (\(y=0\)).
The parabola opens away from the directrix. The vertex is at \(x=2\) and the directrix is at \(x=0\), so the parabola opens to the right.
The distance 'a' from the vertex to the directrix is the horizontal distance between the point (2,0) and the line \(x=0\).
\[ a = |2 - 0| = 2 \]
The focus lies on the axis of symmetry (\(y=0\)) and is at a distance 'a' from the vertex, inside the curve.
Since the parabola opens to the right, the focus will be at \((h+a, k)\).
Focus \(S = (2+2, 0) = (4, 0)\).

Step 4: Final Answer:
The focus of the parabola is at (4, 0).