Find the equation of the chord joining points $p$ and $q$ on the parabola $x=2t$, $y=t^2$ if $p$ and $q$ are the roots of the equation $t^2-4t+2=0$

Question

I have the answer but do not know the process in achieving it. Find the equation of the chord joining points $p$ and $q$ on the parabola $x=2t$, $y=t^2$ if $p$ and $q$ are the roots of the equation $t^2-4t+2=0$.

Answer 1

HINT

Can you obtain the equation of the chord joining $P$ and $Q$ in terms of $p$ and $q$?

You will find the formula involves both $p+q$ and $pq$, which can be read off the quadratic given:

$p+q$ is the sum of roots, therefore 4.

$pq$ is the product of roots, therefore 2

Answer 2

We have $$x=2t, \ y=t^2$$ $$\implies x^2=4y$$

Now, solving $t^2-4t+2=0$, we get $$t=\frac{-(-4)\pm\sqrt{(-4)^2-4(1)(2)}}{2(1)}$$ $$t=\frac{4\pm 2\sqrt 2}{2}=2\pm\sqrt 2$$

Setting the values of $t$, we get the coordinates of the points P & Q as follows $$P(2t, t^2)\equiv (4+ 2\sqrt 2,\ 6+4\sqrt 2 )$$ & $$Q(2t, t^2)\equiv (4- 2\sqrt 2,\ 6-4\sqrt 2 )$$

I hope you can find out the equation of the chord PQ passing through the points $P(4+ 2\sqrt 2,\ 6+4\sqrt 2 )$ & $Q(4- 2\sqrt 2,\ 6-4\sqrt 2 )$ using formula $$y-(6+4\sqrt 2)=\frac{(6+4\sqrt 2)-(6-4\sqrt 2)}{(4+2\sqrt 2)-(4-2\sqrt 2)}(x-(4+ 2\sqrt 2))$$ $$y-(6+4\sqrt 2)=2(x-4-2\sqrt 2)$$ $$y=2x-2$$

$$\bbox[5px, border:2px solid #C0A000]{\color{red}{\text{equation of the chord PQ:}\ \ \color{blue}{2x-y-2=0}}}$$

Answer 3

Use Lagrange's interpolation polynomial:: the chord must pass through the points $(2p,p^2)$ and $(2q,q^2)$, hence the equation is: $$y=p^2\frac{x-2q}{2p-2q}+q^2\frac{x-2p}{2q-2p}=\frac{(p^2-q^2)x-2pq(p-q)}{2(p-q)}=\frac{p+q}2x-pq$$ Using Vieta's relations: $p+q=4,\ pq=2$, it gives the equation: $$y=2(x-1).$$