Parametrics question (1 Viewer)

TeheeCat · Oct 20, 2018

P and Q are the points with parameters p and q on the parabola x = at^2, y = 2at.

a) Show that the chord PQ is 2x - (p+q)y + 2apq = 0

b) If OP is perpendicular to OQ, show that the x-intercept of PQ is independent of p and q.

I need help with both, but I'm particularly confused with b. What does it mean by "independent of p and q"?

Even hints are enough for me!

Thank you!

fan96 · Oct 20, 2018

We have the points $P(ap^2,\,2ap)$ and $Q(aq^2,\,2aq)$

a) The equation of a line joining two points may be found by calculating the gradient (i.e. rise/run) using the two points, and then using the point-gradient formula $(y-y_1) = m(x-x_1)$ with either point.

b) If two lines are perpendicular, the product of their gradients must be $-1$ .

In this case, $m_{OP} \cdot m_{OQ} = -1$ . This condition can be applied to your answer in a).

If a quantity is independent of a variable it means that the value of the variable (whatever it may be) has no influence on what the quantity is.

For example, if $k = 3$ then $k$ is independent of $x$ .

But, if $k = 3x$ then $k$ is NOT independent of $x$ .

HeroWise · Oct 21, 2018

Danneo · Oct 29, 2018

Thx for your working out

, tho i lost track of whats going on in part ii) can you help me understand?

fan96 · Oct 29, 2018

$m_{OQ} = ap^2/2ap = p/2$ and, similarly, $m_{OP} = aq^2/2aq = q/2$ .

$\therefore m_{OQ}\cdot m_{OP} = -1 \implies pq = -4 \quad (*)$

The $x$ intercept of $PQ$ can be found by setting $y=0$ in the equation of $PQ$ . This gives:

$\begin{aligned} x &= -apq \\ &= -a(-4) \quad {\rm from\, (*)} \\ &= 4a \end{aligned}$

Which is independent of either parameter.

Parametrics question (1 Viewer)

TeheeCat

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fan96

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HeroWise

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Danneo

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fan96

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