Chapter 8.3, Problem 32E

To determine

To find: The number of points that passes in parabola

Answer to Problem 32E

There would be infinite number of parabolas passing through the two points.

Explanation of Solution

Given information:

To draw the parabola through the two points with different x-coordinates.

Formula used:

  $f\left(x\right)=a{x}^2+bx+c$

Calculation:

Equation of a parabola is

  $f\left(x\right)=a{x}^2+bx+c$

A parabola passing through 2 points give two linear equations with three variables that need to be satisfied.

Let $\left(x_0,y_0\right)$ and $\left(x_1,y_1\right)$ be two points through which parabola passes.

This gives

  $\begin{array}{l}a{x}_0^2+b{x}_0=y_0-c\\ a{x}_1^2+b{x}_1=y_1-c\end{array}$

Consider two cases:

Case 1: $x_0=0,\text{Then}{x}_1\ne 0.$

Solution would be $c=y_0\text{and}a=\frac{y_1-y_0-b{x}_1}{x_1^2},$ where b can be chosen arbitrarily.

Case 2: Let none of the abscissa is zero:

Note that slope of the lines are not same as different m-coordinates are given. Considering c as given, we know that solution of a and b exists as function of c if the slope of the lines are not same.

As a result, we have infinite solutions of the system of linear equations.

Thus, there would be infinite number of parabolas passing through the two points.

Conclusion:

There would be infinite number of parabolas passing through the two points.