Find the value of $k$ if $4 x + y + k = 0$ is a tangent to the ellipse $x^{2} + 3 y^{2} = 3$

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Solution

Given line and ellipse can be written as $y = - 4 x - k$ and $\frac{x^{2}}{3} + \frac{y^{2}}{1} = 1$
Now using condition of tangency, $c^{2} = a^{2} m^{2} + b^{2}$
$\Rightarrow (- k)^{2} = 3 (- 4)^{2} + 1^{2}$
$\Rightarrow k^{2} = 49$
$∴ k = \pm 7$

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