Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (0, - 25); Directrix: y = 25

검증된 단계별 안내

Recall that the standard form of a parabola with a vertical axis of symmetry is given by the equation \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus (or directrix).

Identify the given focus \( (0, -25) \) and directrix \( y = 25 \). The vertex lies exactly halfway between the focus and the directrix along the vertical line.

Calculate the vertex \( (h, k) \) by finding the midpoint between the focus and directrix: \( k = \frac{-25 + 25}{2} = 0 \), so the vertex is at \( (0, 0) \).

Determine the value of \( p \), which is the distance from the vertex to the focus (or directrix). Since the vertex is at \( y=0 \) and the focus is at \( y=-25 \), \( p = -25 \) (negative because the parabola opens downward).

Substitute \( h=0 \), \( k=0 \), and \( p=-25 \) into the standard form equation to get \( (x - 0)^2 = 4(-25)(y - 0) \), which simplifies to \( x^2 = -100y \). This is the standard form of the parabola.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Understanding this definition helps in deriving the equation of the parabola based on the given focus and directrix.

Standard Form of a Parabola

The standard form of a parabola's equation depends on its orientation. For a vertical parabola with vertex at (h, k), the form is (x - h)^2 = 4p(y - k), where p is the distance from the vertex to the focus or directrix. This form is essential for writing the equation once the vertex and p are known.

Finding the Vertex from Focus and Directrix

The vertex of a parabola lies midway between the focus and the directrix. By calculating the midpoint between the focus (0, -25) and the directrix y = 25, you can find the vertex, which is crucial for expressing the parabola's equation in standard form.