The equation of a tangent to the hyperbola $3 x^{2} - y^{2} = 3$ , parallel to the line $y = 2 x + 4$ is :

y = 2 x + 3

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y = 2 x + 1

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y = 2 x - 1

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y = 2 x + 2

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Solution

The correct options are
B $y = 2 x + 1$
C $y = 2 x - 1$
Equation of hyperbola is
$3 x^{2} - y^{2} = 3$ or
$\frac{x^{2}}{1} - \frac{y^{2}}{3} = 1$
So that , $a^{2} = 1, b^{2} = 3$
Equation of line is $y = 2 x + 4$ , slope of line $= 2$
$∴$ , The tangents are parallel to this line
Slope of tangents $= 2$
$m = 2$
Equation of tangents are
$y = 2 x \pm \sqrt{1 \times 4 - 3}$
$y = 2 x \pm 1$
$y = 2 x + 1, y = 2 x - 1$
Hence , Option B and C.

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