ax - y + c = 0 is the equation of the common | vedclass

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(D) The equation of the tangent to the parabola $y^2 = 8\sqrt{5}x$ is $y = ax + \frac{2\sqrt{5}}{a}$.Comparing this with $y = ax + c$,we get $c = \fra

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$20$

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(D) The equation of the tangent to the parabola $y^2 = 8\sqrt{5}x$ is $y = ax + \frac{2\sqrt{5}}{a}$.Comparing this with $y = ax + c$,we get $c = \frac{2\sqrt{5}}{a}$.The condition for the line $y = ax + c$ to be a tangent to the circle $x^2 + y^2 = 1$ is $c^2 = 1 + a^2$.Substituting $c = \frac{2\sqrt{5}}{a}$ into the condition,we get $(\frac{2\sqrt{5}}{a})^2 = 1 + a^2$.$\frac{20}{a^2} = 1 + a^2$ $\Rightarrow 20 = a^2 + a^4$ $\Rightarrow a^4 + a^2 - 20 = 0$.Let $t = a^2$,then $t^2 + t - 20 = 0$.$(t + 5)(t - 4) = 0$. Since $a^2 > 0$,we have $t = 4$,so $a^2 = 4$.Then $c^2 = 1 + a^2 = 1 + 4 = 5$.Therefore,$a^2c^2 = 4 	imes 5 = 20$.

$ax - y + c = 0$ is the equation of the common tangent to the parabola $y^2 = 8\sqrt{5}x$ and the circle $x^2 + y^2 = 1$. If this tangent makes an acute angle with the positive $X$-axis,then $a^2c^2 =$

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