Mathematics - Hyperbola Question with Solution | TestHub

Let $P\left(t,\frac{-1}{t}\right)$ be any point on $xy=-1$. The equation of the tangent to $xy=-1$ at $P\left(t,\frac{-1}{t}\right)$ is:

$$\frac{x{y}_1+y{x}_1}{2}=-1$$

$$\frac{-x}{t}+yt=-2$$

$$y=\frac{x}{t^2}+\left(\frac{-2}{t}\right)$$

If this tangent is also tangent to the parabola $y^2=8x$, then we can use the condition for tangency. For a parabola $y^2=4ax$, the tangent is $y=mx+\frac{a}{m}$. Here, $4a=8$, so $a=2$.

Comparing $y=\frac{1}{t^2}x-\frac{2}{t}$ with $y=mx+\frac{a}{m}$, we have $m=\frac{1}{t^2}$ and $\frac{a}{m}=-\frac{2}{t}$.

Substituting $a=2$:

$$\frac{2}{1/t^2}=-\frac{2}{t}$$

$$2t^2=-\frac{2}{t}$$

$$t^3=-1$$

Thus, $t=-1$.

Therefore, the common tangent is $y=x+2$.$\mathrm{y}=\mathrm{x}+2$