Let P be the point of intersection of the common tangents to the parabola y^2=12x and hype…

Question

Let $P$ be the point of intersection of the common tangents to the parabola $y^2=12x$ and hyperbola $8x^2-y^2=8$. If $S$ and $S'$ denote the foci of the hyperbola, where $S$ lies on the positive $x$-axis, then $P$ divides $SS'$ in a ratio:

Accepted Answer

Parabola: $a=3$, tangent $y=mx+\dfrac3m$. Hyperbola $x^2-\dfrac{y^2}8=1$: tangent $y=mx+c$, $c^2=m^2-8$.

$$\left(\frac3m\right)^2=m^2-8\Rightarrow9=m^4-8m^2\Rightarrow(m^2-9)(m^2+1)=0\Rightarrow m^2=9.$$

Tangents: $y=3x+1$ and $y=-3x-1$; intersection $P=\left(-\dfrac13,0\right)$.

$S=(3,0)$, $S'=(-3,0)$: $S'P=\dfrac83$, $PS=\dfrac{10}3$, so $P$ divides $SS'$ in ratio $SP:PS'=10:8=5:4$.

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