Finding the Coefficients of a Parabolic Curve: A Comprehensive Guide

In this article, we delve into the process of finding the coefficients of a quadratic curve that satisfies specific geometric constraints. Specifically, we will explore the equation y ax^2 bx c and determine the values of a, b, and c when the curve passes through a given point and is tangent to a line at a particular point.

Introduction

Understanding how to derive the coefficients of a quadratic equation based on given conditions is crucial in both theoretical and practical applications. This article outlines the method to find the coefficients a, b, and c using a step-by-step approach.

Problem Statement

We are given the equation of a parabolic curve: y ax^2 bx c This curve passes through the point (1, 2) and is tangent to the line y x at the origin (0, 0).

Solution

Step 1: Condition at the Point (1, 2)

Since the curve passes through the point (1, 2), we can substitute (x 1) and (y 2) into the equation:

a(1)^2 b(1) c 2 This simplifies to: 1a 1b 1c 2 or simply a b c 2

Step 2: Tangent Condition at the Origin (0, 0)

For the curve to be tangent to the line y x at the origin, two conditions must be met:

The curve must pass through the origin, giving us: a(0)^2 b(0) c 0 which implies c 0 The slope of the curve at the origin must match the slope of the line y x, which is 1. The slope of the curve is given by the derivative: frac{dy}{dx} 2ax b Evaluating this at (x 0) gives 2a(0) b b Therefore, b 1

With c 0 and b 1, we substitute these values back into the equation from Step 1:

a 1 0 2 This simplifies to: a 1 2 which implies a 1

Conclusion

Thus, the coefficients are:

a 1 b 1 c 0

Therefore, the equation of the curve is:

y x^2 x

Final Equation

The final form of the quadratic curve is:

y x^2 x

Summary of Key Steps

Substitute the point (1, 2) into the equation to get a linear equation in terms of a, b, c. Equate the curve to the line at the origin and take the derivative to get the slope condition. Solve the system of equations to find the values of a, b, c.

Practical Applications

The method discussed here can be applied to various problems, such as fitting a curve to given data points or modeling physical phenomena where parabolic or quadratic relationships are relevant.

Related Topics

Geometric Constraints in Curve Fitting Quadratic Equations in Geometry Tangent Lines to Curves