Step-by-step Solution:

1. The given line is rewritten in slope-intercept form: \[ y = -\frac{3}{5}x + \frac{k}{5} \] So, the slope \( m = -\frac{3}{5} \) and the y-intercept is \( c = \frac{k}{5} \). \[\] 2. The ellipse equation is rewritten in standard form: \[ \frac{x^2}{25} + \frac{y^2}{16} = 1 \] Here, \( a^2 = 25 \) and \( b^2 = 16 \). \[\] 3. The condition for tangency states that for a line \( y = mx + c \) to be tangent to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the relation: \[ c^2 = a^2 m^2 + b^2 \] must hold. \[\] 4. Substituting the values: \[ \left(\frac{k}{5}\right)^2 = 25 \times \left(\frac{3}{5}\right)^2 + 16 \] \[ \frac{k^2}{25} = 25 \times \frac{9}{25} + 16 \] \[ \frac{k^2}{25} = 9 + 16 = 25 \] \[ k^2 = 25 \times 25 = 625 \] \[ k = \pm 25 \] Thus, the correct value of \( k \) is \( \pm 25 \).