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Find the equation of the parabola with vertex at (2, 3) and focus at (2, 5).

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Question: Find the equation of the parabola with vertex at (2, 3) and focus at (2, 5).

Options:

  1. y = (1/4)(x - 2)^2 + 3
  2. y = (1/4)(x - 2)^2 - 3
  3. y = (1/4)(x + 2)^2 + 3
  4. y = (1/4)(x + 2)^2 - 3

Correct Answer: y = (1/4)(x - 2)^2 + 3

Solution: 

The vertex form of a parabola is given by (x - h)^2 = 4p(y - k). Here, h = 2, k = 3, and p = 1 (distance from vertex to focus). Thus, the equation is (x - 2)^2 = 4(1)(y - 3) or y = (1/4)(x - 2)^2 + 3.

Find the equation of the parabola with vertex at (2, 3) and focus at (2, 5).

Practice Questions

Q1
Find the equation of the parabola with vertex at (2, 3) and focus at (2, 5).
  1. y = (1/4)(x - 2)^2 + 3
  2. y = (1/4)(x - 2)^2 - 3
  3. y = (1/4)(x + 2)^2 + 3
  4. y = (1/4)(x + 2)^2 - 3

Questions & Step-by-Step Solutions

Find the equation of the parabola with vertex at (2, 3) and focus at (2, 5).
  • Step 1: Identify the vertex of the parabola. The vertex is given as (2, 3). Here, h = 2 and k = 3.
  • Step 2: Identify the focus of the parabola. The focus is given as (2, 5).
  • Step 3: Determine the direction of the parabola. Since the focus (2, 5) is above the vertex (2, 3), the parabola opens upwards.
  • Step 4: Calculate the distance p from the vertex to the focus. The distance p is the difference in the y-coordinates: 5 - 3 = 2.
  • Step 5: Write the vertex form of the parabola. The vertex form is (x - h)^2 = 4p(y - k).
  • Step 6: Substitute h, k, and p into the vertex form. This gives us (x - 2)^2 = 4(2)(y - 3).
  • Step 7: Simplify the equation. This results in (x - 2)^2 = 8(y - 3).
  • Step 8: Rearrange the equation to express y in terms of x. This gives us y = (1/8)(x - 2)^2 + 3.
  • Vertex Form of a Parabola – Understanding the vertex form equation of a parabola and how to identify the vertex and focus.
  • Distance from Vertex to Focus – Calculating the value of 'p' which represents the distance from the vertex to the focus.
  • Coordinate Geometry – Applying knowledge of coordinates to determine the position of the vertex and focus in the context of a parabola.
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