Exercise 1595Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

Shortest distance from a point to a line

#Lagrange#distance#line

Statement

Find the shortest distance from the origin to the line $2x+y=5$ .

Hint

Minimize $x^2+y^2$ under $2x+y=5$ .

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Lagrange. $2x=2\lambda$ , $2y=\lambda$ , so $y=x/2$ . Constraint gives $2x+x/2=5$ , $x=2$ , $y=1$ .

Distance. $\sqrt{4+1}=\sqrt5$ .

Conclusion. Distance $=\sqrt5$ .