Find the equation of the parabola with vertex at (2, 3) and focus at (2, 5).
Practice Questions
1 question
Q1
Find the equation of the parabola with vertex at (2, 3) and focus at (2, 5).
y = (1/4)(x - 2)^2 + 3
y = (1/4)(x - 2)^2 - 3
y = (1/4)(x + 2)^2 + 3
y = (1/4)(x + 2)^2 - 3
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.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the equation of the parabola with vertex at (2, 3) and focus at (2, 5).
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.
Steps: 8
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.
