Question

The equation of the parabola with focus (2, 0) and vertex (1, 0) is

• A. \(y^2 = 4x\)
• B. \(y^2 = 4x - 4\)
• C. \(y^2 = 4(x+1)\)
• D. \(y^2 = -4(x-1)\)

Hint:

The position of the focus relative to the vertex determines the direction the parabola opens. If the focus is to the right of the vertex, it opens right. If left, it opens left. If above, it opens up. If below, it opens down. This directly tells you which standard form of the equation to use.

Answer 1

The Correct Option is B

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

Step 2: Key Formula or Approach:
First, determine the orientation and the parameter 'a' of the parabola.
The vertex is \((h, k)\). The focus is \((h+a, k)\) for a parabola opening right.
The standard equation for a parabola opening horizontally with vertex at \((h, k)\) is:
\[ (y-k)^2 = 4a(x-h) \]

Step 3: Detailed Explanation:
The vertex is given as \(V(h, k) = (1, 0)\).
The focus is given as \(S = (2, 0)\).
Since the y-coordinates of the vertex and focus are the same, the axis of symmetry is horizontal (the x-axis, y=0).
The focus (x=2) is to the right of the vertex (x=1), so the parabola opens to the right.
The distance 'a' is the distance between the vertex and the focus.
\[ a = \sqrt{(2-1)^2 + (0-0)^2} = \sqrt{1^2} = 1 \]
Now we use the standard equation \((y-k)^2 = 4a(x-h)\) with \(h=1, k=0, a=1\).
\[ (y-0)^2 = 4(1)(x-1) \]
\[ y^2 = 4(x-1) \]
Expanding the equation, we get:
\[ y^2 = 4x - 4 \]

Step 4: Final Answer:
The equation of the parabola is \(y^2 = 4x - 4\).