Let a line L: 2x + y = k, k 0 be a tangent | vedclass

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(C) The equation of the line is $y = -2x + k$. Since it is a tangent to the hyperbola $x^2 - y^2 = 3$ (where $a^2 = 3, b^2 = 3$),the condition for tan

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$-24$

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(C) The equation of the line is $y = -2x + k$. Since it is a tangent to the hyperbola $x^2 - y^2 = 3$ (where $a^2 = 3, b^2 = 3$),the condition for tangency $y = mx \pm \sqrt{a^2m^2 - b^2}$ gives:$k = \sqrt{3(-2)^2 - 3} = \sqrt{3(4) - 3} = \sqrt{9} = 3$ (since $k > 0$).So,the line is $y = -2x + 3$.For this line to be a tangent to the parabola $y^2 = \alpha x$,it must satisfy the condition $c = \frac{a}{m}$,where $y = mx + c$ and $a$ in the parabola equation $y^2 = 4Ax$ is $\frac{\alpha}{4}$.Here,$m = -2$ and $c = 3$. The condition for tangency to $y^2 = \alpha x$ is $c = \frac{\alpha}{4m}$.Substituting the values: $3 = \frac{\alpha}{4(-2)} = \frac{\alpha}{-8}$.Therefore,$\alpha = 3 	imes (-8) = -24$.

Let a line $L: 2x + y = k, k > 0$ be a tangent to the hyperbola $x^2 - y^2 = 3$. If $L$ is also a tangent to the parabola $y^2 = \alpha x$,then $\alpha$ is equal to:

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