# Parametrics question (1 Viewer)

#### TeheeCat

##### Active Member
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

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

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 $\bg_white (y-y_1) = m(x-x_1)$ with either point.

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

In this case, $\bg_white m_{OP} \cdot m_{OQ} = -1$. 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 $\bg_white k = 3$ then $\bg_white k$ is independent of $\bg_white x$.

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

#### Danneo

##### Member
Thx for your working out , tho i lost track of whats going on in part ii) can you help me understand?

#### fan96

##### 617 pages
$\bg_white m_{OQ} = ap^2/2ap = p/2$ and, similarly, $\bg_white m_{OP} = aq^2/2aq = q/2$.

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

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

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

Which is independent of either parameter.