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:

• A. 25
• B. 30
• C. 15
• D. 10

Answer 1

The Correct Option is A

The equation of the line is:

\[ y + 9 = m(x - 4). \]

Find \( A \) and \( B \) (intersection points with the axes):

At \( y = 0 \), \( x = \frac{9}{m} + 4 \implies A\left(\frac{9}{m} + 4, 0\right) \).

At \( x = 0 \), \( y = -9 - 4m \implies B(0, -9 - 4m) \).

The sum of distances:

\[ OA + OB = \sqrt{\left(\frac{9}{m} + 4\right)^2} + \sqrt{(-9 - 4m)^2}. \]

Using AM-GM inequality, the minimum value occurs when:

\[ m = \frac{3}{2}. \]

Substitute \( m \) to get:

\[ OA + OB = 25. \]