# Help with math extension 2 question, thanks! (1 Viewer)

#### creativemaster

##### New Member
Let H and K be the points representing the roots of
x^2 +2px+q=0, where p and q are real and p^2 <q . Find the algebraic relation satisfied by p and q in each of the following cases
(i) angle HOK is a right angle.
(ii) A, B, H and K are equidistant from the origin.

#### fan96

##### 617 pages
This question should go in the Maths subforum.

$\bg_white x = -p \pm \sqrt{p^2-q}$

Because $\bg_white p^2 - q < 0$, we can use the identity $\bg_white \sqrt{ab} = \sqrt a \sqrt b$ with $\bg_white a = -1$ and $\bg_white b = q - p^2$.

(Note that this identity is not valid if both $\bg_white a$ and $\bg_white b$ are negative)

Hence,

$\bg_white x = -p \pm i\sqrt{q-p^2}$

Where $\bg_white p$ and $\bg_white \sqrt{q-p^2}$ are both real numbers.

Therefore, the gradient of the line joining the root to the origin in the complex plane is given by

$\bg_white m = \pm \frac{\sqrt{q-p^2}}{p}$

And recall that the requirement for two lines to be perpendicular is that their gradients multiply to be $\bg_white -1$.

The rest should be easy.