Question

Find the x-coordinates of the two points on the curve

Answer 1

Answer: x = {-2, 2}

Step-by-step explanation:

Tangent means it is touching. Find the intersection of the two equations.

Solve the linear equation for y, then set the two equations equal to each other.

$4y=x+8\qquad \rightarrow \qquad y=\dfrac{x+8}{4}$

$\dfrac{x-1}{x}=\dfrac{x+8}{4}\\\\\\\text{Cross multiply and solve for x:}\\4(x-1)=x(x+8)\\4x-4=x^2+8x\\.\qquad 0=x^2+4x+4\\.\qquad 0=(x+2)^2\\.\qquad 0=x+2\\.\qquad x=-2$

To find the next point that is parallel to the linear equation and tangent to the curve, we need to use the linear equation with slope (m) = $\dfrac{1}{4}$ and unknown b.

Let's try b = 0, then the equation of the linear equation is: $y=\dfrac{1}{4}x$

Set the equations equal to each other and solve for x:

$\dfrac{x-1}{x}=\dfrac{x}{4}\\\\\\4(x-1)=x^2\\4x-4=x^2\\.\qquad 0=x^2-4x+4\\.\qquad 0=(x-2)^2\\.\qquad 0=x-2\\.\qquad x=2$

This works!!!  If it didn't work, we would have tried other values for b until we arrived at a solution.