Chapter 9.3, Problem 38WE

To determine

To explain: The line joining the two given points has y-intercepts $ab$ .

Explanation of Solution

Given information: The line passes through $\left(-a,a^2\right)\text{and}\left(b,b^2\right)$ .

Formula used: Point −slope formula: $y-y{1}_{}=m\left(x_2-x_1\right)$

Proof: Substitute the given coordinates into the formula, $y-y{1}_{}=m\left(x_2-x_1\right)$

  $\begin{array}{l}y=x_2+\left(\frac{y_2-y_1}{x_2-x_1}\right)\left(x_2-x_1\right)\\ y=a^2+\frac{b^2-a^2}{b+a}\left(x+a\right)\\ y=a^2+\left(b-a\right)\left(x+a\right)\end{array}$

As the intersection with the y-axis, so $y\left(0\right)$ ,

  $\begin{array}{l}y=a^2+\left(b-a\right)\left(x+a\right)\\ y\left(0\right)=a^2+\left(b-a\right)\left(0+a\right)\\ y\left(0\right)=a^2+\left(b-a\right)a\\ y\left(0\right)=a^2+ab-a^2\\ \therefore y\left(0\right)=ab\end{array}$

Hence, proved.