Parametric Question for old New Senior Mathematics 3U (1 Viewer)

McRafi9 · Jan 19, 2017

Ex24(c) Q11. pg.79

P is a variable point on the parabola x^2 = -4y. The tangent from P cuts the parabola x^2 = 4y at Q and R. Show that 3x^2 = 4y is the equation of the locus of the mid-point of the chord RQ.

InteGrand · Jan 19, 2017

McRafi9 said:
Ex24(c) Q11. pg.79

P is a variable point on the parabola x^2 = -4y. The tangent from P cuts the parabola x^2 = 4y at Q and R. Show that 3x^2 = 4y is the equation of the locus of the mid-point of the chord RQ.

$\noindent Let $P$ be parametrised as $\left(-2t, -t^{2}\right)$ ($t\in \mathbb{R}$). The tangent at $P$ has slope $t$, so has equation $y = -t^{2} + t\left(x + 2t\right)$, i.e. $y = tx +t^{2}$. The intersection points with the tangent and the parabola $x^{2} = 4y$ thus satisfy $x^{2} = 4\left(tx + t^{2}\right) \iff x^{2} -4tx -4t^{2} = 0$. The sum of roots is $4t$, so the midpoint $M$ of $RQ$ has $x$-value $x(t) = 2t$ (half the sum of roots). For the $y$-value of the midpoint, it is $y(t) = t\cdot 2t + t^{2} = 3t^{2}$, since $M$ lies on the line $y= tx + t^{2}$ (tangent line we found earlier). Thus the coordinates of $M$ are $x(t) = 2t$, $y(t) = 3t^{2}$. Hence $3x(t)^{2} = 3\times 4t^{2} = 4y(t)$, which proves the desired result.$

McRafi9 · Jan 19, 2017

InteGrand said:
$\noindent Let $P$ be parametrised as $\left(-2t, -t^{2}\right)$ ($t\in \mathbb{R}$). The tangent at $P$ has slope $t$, so has equation $y = -t^{2} + t\left(x + 2t\right)$, i.e. $y = tx +t^{2}$. The intersection points with the tangent and the parabola $x^{2} = 4y$ thus satisfy $x^{2} = 4\left(tx + t^{2}\right) \iff x^{2} -4tx -4t^{2} = 0$. The sum of roots is $4t$, so the midpoint $M$ of $RQ$ has $x$-value $x(t) = 2t$ (half the sum of roots). For the $y$-value of the midpoint, it is $y(t) = t\cdot 2t + t^{2} = 3t^{2}$, since $M$ lies on the line $y= tx + t^{2}$ (tangent line we found earlier). Thus the coordinates of $M$ are $x(t) = 2t$, $y(t) = 3t^{2}$. Hence $3x(t)^{2} = 3\times 4t^{2} = 4y(t)$, which proves the desired result.$

Wow. Just wow. Thanks.

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Parametric Question for old New Senior Mathematics 3U (1 Viewer)

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