Animated Solution for Mathematics - Conic Sections: The equation of the common tangent to the curves $y^2=8x$ and $xy=-1$ is

We are tasked with finding the common tangent to the parabola $y^2=8x$ and the rectangular hyperbola $xy=-1$.

The parabola is defined by $y^2=8x$. Comparing this to the standard form $y^2=4ax$, we identify $4a=8$, which yields $a=2$.

Any tangent to this parabola can be expressed in terms of its slope $m$ as:

Substituting $a=2$, the equation of the tangent becomes:

For this line to also be a tangent to the hyperbola $xy=-1$, it must intersect the hyperbola at exactly one point. We substitute the expression for $y$ into the hyperbola's equation:

Expanding this expression, we obtain:

To eliminate the fraction, we multiply the entire equation by $m$:

For the line to be a tangent, the quadratic equation $m^2{x}^2+2x+m=0$ must have equal roots. This occurs when the discriminant $D=B^2-4AC$ is equal to zero.

Here, $A=m^2$, $B=2$, and $C=m$. Setting the discriminant to zero:

Dividing by 4, we arrive at $1-m^3=0$, which implies $m^3=1$. The only real solution for the slope is:

We now substitute the slope $m=1$ back into our tangent equation $y=mx+\frac{2}{m}$.

The equation of the common tangent is: