Let a variable line passing through the centre of the circle x^2+y^2-16x-4y=0 meet the pos…

Question

Let a variable line passing through the centre of the circle $x^2+y^2-16x-4y=0$ meet the positive coordinate axes at the points $A$ and $B$. Then the minimum value of $OA+OB$, where $O$ is the origin, is equal to:

Accepted Answer

Centre $(8,2)$. Line $\dfrac{x}{a}+\dfrac{y}{b}=1$ through $(8,2)$: $\dfrac8a+\dfrac2b=1$.

By Cauchy--Schwarz: $(a+b)\left(\dfrac8a+\dfrac2b\right)\ge(\sqrt8+\sqrt2)^2=18$, so $a+b\ge18$ (equality attainable).

Solutions · 1