Finding the Standard Equation, Axis of Symmetry, and Latus Rectum of a Parabola
Introduction
In the study of conic sections, parabolas are a fascinating topic. This article delves into the process of determining the standard equation, axis of symmetry, and latus rectum of a parabola given its vertex and focus. By understanding these concepts, students and enthusiasts can better comprehend the geometric properties of parabolas. Let's explore the problem statement and the step-by-step solution method.Problem Statement
Consider a parabola with the vertex at (-32, -22) and the focus at (-22, -22). The task is to find the standard equation of the parabola, the axis of symmetry, and the length of the latus rectum.Step-by-Step Solution
1. **Shifting the Vertex to the Origin** To simplify the problem, we shift the vertex to the origin. This is done by transforming the coordinates of the vertex and the focus. The original coordinates are: - Vertex: (-32, -22) - Focus: (-22, -22) To shift the vertex to the origin, we add 32 to both the x and y coordinates of the vertex and the focus. The new coordinates become: - Transformed Vertex: (0, 0) - Transformed Focus: (10, 0) 2. **Equation of the Parabola in the New System** In the new coordinate system where the vertex is at the origin (0, 0) and the focus is at (10, 0), the parabola is described by the standard form equation:[(x - h)^2 4p(y - k)]
Where (h, k) are the coordinates of the vertex and p is the distance from the vertex to the focus.
In our transformed system, the vertex (h, k) is (0, 0), and the distance p from the vertex to the focus is 10. Therefore, the equation of the parabola in the new system is:x^2 40y
3. **Finding the Axis of Symmetry** The axis of symmetry of a parabola in the standard form x^2 4py is the line y k, which passes through the vertex (0, 0) in our new system. Therefore, the axis of symmetry in the transformed system is y 0 or the x-axis. 4. **Length of the Latus Rectum** The length of the latus rectum of a parabola is given by 4p. In our case, p 10, so the length of the latus rectum is 40. 5. **Transforming Back to the Original System** To express the standard equation, axis of symmetry, and latus rectum in the original coordinate system, we reverse the transformation. This means we replace x with x - 32 and y with y - 22. - **Standard Equation**[(x - 32)^2 40(y - 22)]
- **Axis of Symmetry**The x-axis in the transformed system corresponds to the line y 22 in the original system.
- **Length of Latus Rectum**The length remains the same: 40.