The value of $c$ for which the line $y = 3 x + c$ touches the ellipse $16 x^{2} + y^{2} = 16$ is:

5

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1

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4

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Solution

The correct option is A $5$
Given $y = 3 x + c$ and $16 x^{2} + y^{2} = 16$

Put $y = 3 x + c$ in $16 x^{2} + y^{2} = 16$

$\Rightarrow 16 x^{2} + (3 x + c)^{2} = 16$

$\Rightarrow 25 x^{2} + 6 c x + c^{2} = 16 . . (1)$

So for given line to touch the ellipse discriminant of quadratic equation (1) has to be zero as it will touch at a single point

$\Rightarrow (6 c)^{2} - 4 \times 25 \times (c^{2} - 16) = 0$

$\Rightarrow 36 c^{2} - 100 c^{2} - 1600 = 0$

$\Rightarrow 64 c^{2} = 1600$

$\Rightarrow c^{2} = 25 \Rightarrow c = \pm 5$

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