"Let a line L : 2x + y = k, k 0 be a tangent to the hyperbola x2 $-$ y2 = 3. If L is also a tangent to the parabola y2 = $\alpha$x, then $\alpha$ is equal to :"

Question

"Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 $-$ y2 = 3. If L is also a tangent to the parabola y2 = $\alpha$x, then $\alpha$ is equal to :"

Accepted Answer

"Tangent to hyperbola of

Slope m = $-$2 (given)

y = $-$2x $\pm$ $\sqrt {3(3)}$

$\left( {y = mx \pm \sqrt {{a^2}{m^2} - {b^2}} } \right)$

$\Rightarrow$ y + 2x = $\pm$ 3 $\Rightarrow$ 2x + y = 3 (k > 0)

For parabola y² = $\alpha$x

$y = mx + {\alpha \over {4m}}$

$\Rightarrow y = - 2x + {\alpha \over { - 8}}$

$\Rightarrow {\alpha \over { - 8}} = 3$

$\Rightarrow \alpha = - 24$"

Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x² $-$ y² = 3. If L is also a tangent to the parabola y² = $\alpha$x, then $\alpha$ is equal to :

Solution

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Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 $-$ y2 = 3. If L is also a tangent to the parabola y2 = $\alpha$x, then $\alpha$ is equal to :

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Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x² $-$ y² = 3. If L is also a tangent to the parabola y² = $\alpha$x, then $\alpha$ is equal to :