Let a variable line of slope m0 passing through the point (4,-9) intersect the coordinate…

Question

Let a variable line of slope $m>0$ passing through the point $(4,-9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is

Accepted Answer

$$\begin{aligned}
& L:(y+9)=m(x-4) \
& A:\left(4+\frac{9}{m}, 0\right) \
& B:(0,-9-4 m) \
& O A+O B]_{\min } \
& \Rightarrow E_{\min }=4+\frac{9}{m}+9+4 m \
& E=13+\frac{9}{m}+4 m \
& \frac{d E}{d M}=0 \Rightarrow-\frac{9}{m^2}+4=0 \Rightarrow m= \pm \frac{3}{2}
\end{aligned}$$

$$\begin{aligned}
& \frac{d^2 E}{d M^2}=\frac{18}{m^3}>0 \text { for } \quad m=\frac{3}{2} \
& \therefore E_{\min }=4+6+9+6=25
\end{aligned}$$

Solutions · 1