Circles
Minimum of OA+OB for Variable Line Through Fixed Point
nta_pyq_2024_jan
Grade 11
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
Step-by-Step Solution
Key Concept: Centre of circle: $(8,2)$. Line through $(8,2)$ with slope $m$: $y-2=m(x-8)$. x-intercept: $-2/m+8$; y-intercept: $-8m+2$. Minimize $OA+OB=(-2/m+8)+(-8m+2)=10-2/m-8m$. Differentiate w.r.t. $m$ and set $=0$.
OA+OB $=10+2/t+8t\ge10+2\sqrt{2/t\cdot8t}=10+8=18$ (AM-GM). Minimum $=18$ at $t=1/2$.
Correct Answer: 2