Question

Shortest distance between curve and point?

Answer 1

\[y=\frac{ 7 }{ x } , \left( 0, 0 \right)\]

Answer 2

Would I use Pythagorean thm?
\[c ^{2} = x ^{2} + \left( \frac{ 7 }{ x } \right)^{2}\]

Answer 3

shortest line between a point and a curve is perpendicular to the tangent line of the curve

find slope of tangent using derivative
\[\frac{dy}{dx} = -\frac{7}{x^2}\]
thus slope of perpendicular line is:
\[m = \frac{x^2}{7}\]
Now slope of line between origin and any point on curve is:
\[\frac{y}{x} = \frac{\frac{7}{x}}{x} = \frac{7}{x^2}\]

Set these slopes equal to each other and solve for x:
\[\rightarrow \frac{x^2}{7} = \frac{7}{x^2}\]
\[x = \sqrt{7}\]
thus point on curve closest to origin is:
\[(\sqrt{7}, \sqrt{7})\]

Answer 4

oh yes you can also use pythagorean thm to obtain distance function

you then would have to minimize distance by setting derivative equal to 0