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

(1) 25 

(2) 30 

(3) 15 

(4) 10

Answer 1

Correct option is (1) 25   

equation of line is

y + 9 = m (x – 4)

\(\therefore \mathrm{A}=\left(\frac{9+4 \mathrm{~m}}{\mathrm{~m}}, 0\right)\)

\(B=(0,-9-4 m)\)

\(\therefore \mathrm{OA}+\mathrm{OB}=\frac{9+4 \mathrm{~m}}{\mathrm{~m}}+9+4 \mathrm{~m}\)

\(\because \mathrm{m}>0\)

\(=13+\frac{9}{\mathrm{~m}}+4 \mathrm{~m}\)

\(\because \frac{4 \mathrm{~m}+\frac{9}{\mathrm{~m}}}{2} \geq \sqrt{36} \Rightarrow 4 \mathrm{~m}+\frac{9}{\mathrm{~m}} \geq 12\)

\( \therefore \mathrm{OA}+\mathrm{OB} \geq 25\)