Question

Given points \(\left(p_{1}, q_{1}\right),\left(p_{2}, q_{2}\right),\) and \(\left(p_{3}, q_{3}\right)\) in the plane with \(p_{1}, p_{2},\) and \(p_{3}\) distinct, show that they lie on some curve with equation \(y=a+b x+\) \(c x^{2}\).

Short answer

Three distinct points can always lie on a quadratic curve.

Step-by-step solution

Understand the Problem

The problem requires us to prove that three distinct points in a plane can lie on a curve defined by a quadratic equation, which is a parabola. The general form of the quadratic equation is given as \(y = ax^2 + bx + c\), and we need to find values of \(a\), \(b\), and \(c\) that satisfy the equation for all given points.

Set Up the Equations

Substitute the coordinates of the given points into the quadratic equation. For the points \((p_1, q_1)\), \((p_2, q_2)\), and \((p_3, q_3)\), set up the equations as follows: 1. \(q_1 = ap_1^2 + bp_1 + c\)2. \(q_2 = ap_2^2 + bp_2 + c\)3. \(q_3 = ap_3^2 + bp_3 + c\)

Create a System of Equations

The equations from Step 2 form a system of three linear equations:1. \(ap_1^2 + bp_1 + c = q_1\)2. \(ap_2^2 + bp_2 + c = q_2\)3. \(ap_3^2 + bp_3 + c = q_3\)Since \(p_1\), \(p_2\), and \(p_3\) are distinct, this system can be solved for \(a\), \(b\), and \(c\).

Solve the System of Equations

To solve for \(a\), \(b\), and \(c\), you can use one of several methods, such as substitution, elimination, or matrix methods. One effective approach is to employ matrix inversion if computational tools are available or Gaussian elimination.Solve the system to obtain specific numeric values or expressions for \(a\), \(b\), and \(c\).

Conclude the Existence of a Solution

After solving the system, you'll find a set of values for \(a\), \(b\), and \(c\). These values confirm that there is indeed a quadratic curve that passes through all three points. The uniqueness comes from the fact that three distinct points uniquely determine a quadratic curve.

Key concepts

System of Linear Equations

In this context, a system of linear equations consists of several equations that attempt to express the relationship between variables. Here, we are concerned with three points \( (p_1, q_1), (p_2, q_2), (p_3, q_3) \) related through a quadratic equation.

To determine if they lie on the curve defined by \( y = ax^2 + bx + c \), you substitute the coordinates of each point into this equation. This process creates a system of three equations: \(\begin{align*}q_1 & = ap_1^2 + bp_1 + c, \q_2 & = ap_2^2 + bp_2 + c, \q_3 & = ap_3^2 + bp_3 + c. \end{align*}\)

Each equation is linear in terms of \( a, b, \) and \( c \). Solving this system means finding specific values for these variables that make all equations true simultaneously.

Matrix Methods

Matrix methods offer a streamlined way to solve systems of equations, especially when dealing with multiple equations and variables. In the case of our quadratic curve problem, these methods can be highly effective.

We can organize our system of equations using matrix notation. This involves writing the system as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the column matrix of unknowns \( [a, b, c] \), and \( B \) is the constant matrix. The system looks like this:

\[\begin{bmatrix}p_1^2 & p_1 & 1 p_2^2 & p_2 & 1 p_3^2 & p_3 & 1 \\end{bmatrix}\begin{bmatrix}a b c\end{bmatrix} = \begin{bmatrix}q_1 q_2 q_3\end{bmatrix}\]

Solving this system can be done using various techniques such as matrix inversion or Gaussian elimination, both of which can simplify the computation process.

Gaussian Elimination

Gaussian elimination is a method used to solve linear systems by systematically reducing the equations to arrive at a solution.

It involves three main steps:

• Forward Elimination: Use row operations to change the system into an upper triangular matrix. This transforms the equations, making them easier to solve.
• Back Substitution: Once in triangular form, solve for the variables starting from the last equation.
• Solution Interpretation: After obtaining values for \( a, b, c \), you can establish the existence of the quadratic curve passing through the given points.

Gaussian elimination is particularly useful because it applies systematic steps, making it easy to implement manually or via computer algorithms.