If a tangent to the hyperbola xy = -1 is also | vedclass

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(C) Let the equation of the tangent to the parabola $y^2 = 8x$ be $y = mx + \frac{a}{m}$,where $a = 2$. So,$y = mx + \frac{2}{m}$.This line is also a

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$x - y + 2 = 0$

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(C) Let the equation of the tangent to the parabola $y^2 = 8x$ be $y = mx + \frac{a}{m}$,where $a = 2$. So,$y = mx + \frac{2}{m}$.This line is also a tangent to the hyperbola $xy = -1$,which can be written as $y = -\frac{1}{x}$.Substituting $y = mx + \frac{2}{m}$ into $xy = -1$,we get $x(mx + \frac{2}{m}) = -1$,which simplifies to $mx^2 + \frac{2}{m}x + 1 = 0$.Since the line is a tangent,the discriminant of this quadratic equation must be zero.$D = (\frac{2}{m})^2 - 4(m)(1) = 0$.$\frac{4}{m^2} - 4m = 0 \implies 4 = 4m^3 \implies m^3 = 1 \implies m = 1$.Substituting $m = 1$ into the tangent equation $y = mx + \frac{2}{m}$,we get $y = x + 2$,or $x - y + 2 = 0$.

If a tangent to the hyperbola $xy = -1$ is also a tangent to the parabola $y^2 = 8x$,then the equation of that tangent is

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