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 [2024]

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Solution

Q.
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 [2024]

1 10

2 25

3 30

4 15

Ans.
(2)

Equation of line passing through (4, –9) and having slope m is given by

(y + 9) = m(x – 4)

Since it intersect the coordinate axes at points A and B so at A, y = 0

$∴ A = (\frac{9 + 4 m}{m}, 0)$

Similarly at B, x = 0

$∴ B = (0, - 4 m - 9)$

Now, let O(0, 0) be origin, then

$O A + O B = \sqrt{{(\frac{9 + 4 m}{m})}^{2}} + \sqrt{{(9 + 4 m)}^{2}}$

$= \frac{9 + 4 m}{m} + 9 + 4 m$ $[∵ m > 0]$

$= 13 + \frac{9}{m} + 4 m$

Now, $\frac{\frac{9}{m} + 4 m}{2} \geq \sqrt{\frac{9}{m} \cdot 4 m}$ $[∵ A . M . \geq G . M .]$

$\Rightarrow \frac{9}{m} + 4 m \geq 12$

$∴ O A + O B \geq 13 + 12 i . e . 25$

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