$\text{ The number of values of }c\text{ such that the straight line }y=4x+c\text{ touches the curve }\frac{x^2}{4}+\frac{y^2}{1}=1\text{ is }$

$\text{For given slope, there exist two parallel tangents to the ellipse. }$

$\mathrm{Hence}, \mathrm{there} \mathrm{are} \mathrm{two} \mathrm{values} \mathrm{of} c.$