Interesting parabola property

Question

If a family of straight lines can be represented by an equation $$k^2P+kQ+R=0$$ where 'k' is a parameter and P, Q and R are linear functions of x and y then the family of lines will be tangent to the curve $Q^2=PR$.

I found this property about parabolas in my book. I am wondering how to prove this. I tried by assuming a general parabola but am not able to get there. Need some help.

When $k_1\ne k_2,$ let $S(k_1,k_2)$ be the intersection of the lines $k_i^2P+k_iQ+R$ (for $i=1,2$). Fix $k_1$ and let $k_2\to k_1.$ Then $S(k_1,k_2)$ should converge to a point on $Q^2=PR$ where the line $k_1^2P+k_1Q+R$ is tangent to $Q^2+PR.$ — DanielWainfleet, Apr 15 '17 at 00:07

score 2 · Accepted Answer · answered Apr 14 '17 at 17:56

The family of tangents to a curve is called its envelope. Given a set of lines $F(k,P,Q,R) = 0$, to be tangent to the curve, we must also have $\partial F/\partial k (k,P,Q,R) = 0$ as well (consider the curves $F(k,P,Q,R)=0$ and $F(k',P,Q,R)=0$, and let $k'\to k$). In particular, since in this case $F$ is a polynomial, it suffices to look for a double root, which occurs precisely when the discriminant vanishes. This, of course, is $Q^2-4PR=0$, which is a parabola.

Interesting parabola property

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