$\begin{array}{1 1}(A)\;2x+19 \\(B)\;-2x+\sqrt {19} \\(C)\;2x-\sqrt {19} \\(D)\;x+\sqrt {19} \end{array}$

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Equation of ellipse is

$\large\frac{x^2}{4}+\frac{y^2}{3}$$=1$

$a^2=4\;b^2=3$

Slope of the line $y+2x=4$ is $-2$

Now equation of a tangent having slope m is

$y= mx +\sqrt {a^2m^2+b^2}$

If the tangent is parallel to the given line , then

$m=-2$

Equation of the tangent is $y= -2x +\sqrt {19}$

Hence B is the correct answer.

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