calculate the distance from the parabola y = x² + 1 to the line x - y = 2

Accepted Solution

Let (h,k) be the point closest to line. First find the derivative of the parabola: $$y'=2x$$ And slope of the given line is calculated as: $$ x-y=2$$ implies $$y=x-2 $$ comparing with the slope-intercept form we have: $$slope=1 $$ then $$ 2h=1 $$ $$h=\frac{1}{2} $$ and $$k=h^2+1 = \left(\frac{1}{2}\right)^2+1=\frac{1}{4}+1=\frac{5}{4}$$ SO the point is: $$\left(\frac{1}{2}, \frac{5}{4} \right) $$ So the distance of point from the line is: $$\frac{\frac{1}{2}-2-\frac{5}{4} }{ \sqrt{1^1+1^2} }=\frac{-\frac{11}{4}}{\sqrt{2}}=\frac{-11}{4\sqrt{2}} $$