Chapter 4.9, Problem 56PPSE

Solution

To Find: The value of $a$ for which the line $y=x+a$ separates the parabolas $y=x^2-3x+2$

and $y=-x^2+8x-15$ .

The value of $a$ is $-2.75<a<-2$ .

Given:

Given line: $y=x+a$

Given parabolas: $y=x^2-3x+2$ and $y=-x^2+8x-15$ .

Explanation:

Find the value of $a$:

The given two parabolas' do not Intersect. Therefore, a line will separate them

  $\begin{array}{l}y=x^2-3x+2\to \left(1\right)\\ y=-x^2+8x-15\to \left(1\right)\end{array}$

  $y=x+a$ is a line with slope $1$ and $y-$ intercept $a$ .

Find the tangents to each of the two parabola's with slope $1$ :

As a result, all lines between those tangents will be of the form $y=a+x$ and the two parabolas will also be separated by these lines.

Slope of the tangent must be 1, at a point $\frac{dy}{dx}$

  $y=x^2-3x+2$

Differentiate both sides with respect to $x$ ,

  $\frac{dy}{dx}=2x-3$

Since, $\frac{dy}{dx}=1$ :

  $2x-3=1$

It occurs when $x=2$

Substitute $x=2,$ in $y=x^2-3x+2$ :

  $\begin{array}{c}y={\left(2\right)}^2-3\left(2\right)+2\\ =4-6+2\\ =0\end{array}$

Hence, the tangent to parabola $1$ has slope at the point $\left(2,0\right)$ .

Thus, the equation of the tangent is:

  $\begin{array}{c}\frac{y-0}{x-2}=1\\ y=x-2\end{array}$

The slope of the tangent must be $1$ , at a point $\frac{dy}{dx}$ :

  $y=-x^2+8x-15$

Differentiate with respect to

  $x:$

  $\frac{dy}{dx}=-2x+8$

Since, $\frac{dy}{dx}=1$ :

  $-2x+8=1$

Substitute $x=3.5$ in $y=-x^2+8x-15$ ::

  $\begin{array}{c}y=-{\left(3.5\right)}^2+8\left(3.5\right)-15\\ =0.75\end{array}$

Thus, tangent to parabola $2$ has slope at the point $\left(3.5,0.75\right)$

Thus, the Equation of the tangent is"

  $\begin{array}{c}\frac{y-0.75}{x-3.5}=1\\ y=x-2.75\end{array}$

The line $y=mx+b_2$ will lie between $y=mx+b_1$ and $y=mx+b_3$ , if and only if $b_1<b_2<b_3$ or $b_3<b_2<b_1$

Thus, the line $y=x+a$ will separate the two parabolas if:

  $-2.75<a<-2$

The graph below shows parabola $1$ in black and parabola $2$ in blue with slope $=1$

Example of line $y=x-2.5$ is shown as a dotted line in green colour.