Question

Minimize 

F = x² + y²

 with 

x + 2y = 10.

Answer 1

If the given line is a tangent to the circle x²+y²=F, the radius is √F. The slope of the given line is found by rewriting the given equation of the line y=5-½x, so the gradient is -½, and the gradient of the perpendicular radius is therefore 2. The radius has to pass through the origin so the equation of the radius is y=2x.

This line meets the given line when 2x=5-½x, from which x=2 and y=4. Therefore F=4+16=20. This is the minimum value of F, because if the radius were shorter it would not reach the line.