# Locus and the Parabola Problem

• Nov 29th 2011, 09:39 PM
eskimogenius
Locus and the Parabola Problem
Hi there MHF,

After an hour of intensive algebra, I am still unable to solve this question.

Could someone please help me with Part A). And also, if they are kind enough, Part B).

Thank you.

http://img545.imageshack.us/img545/858/locus.png
• Nov 30th 2011, 12:25 AM
earboth
Re: Locus and the Parabola Problem
Quote:

Originally Posted by eskimogenius
Hi there MHF,

After an hour of intensive algebra, I am still unable to solve this question.

Could someone please help me with Part A). And also, if they are kind enough, Part B).

Thank you.

1. You probably have determined the coordinates of the focus as S(0, 1).

2. The equation of the parabola is $y=\frac14 \cdot x^2$.

That means the slope of the tangent to the parabola is calculated by $y'=\frac12 \cdot x$

3. Let $P\left(p,\frac14p^2 \right)$ be a point on the parabola. The tangent $t_P$ at P to the parabola has the equation:

$y-\frac14p^2=\frac12p \cdot (x-p)$

$t_P: y=\frac12p \cdot x - \frac14p^2$

4. The normal $n_P$ in P has the slope $m = -\frac2p$. Since the normal passes through P too it has the equation:

$y-\frac14p^2=-\frac2p \cdot (x-p)$

$y = -\frac2p \cdot x +2+\frac14p^2$

5. Now determine the point of intersection of the parabola and the normal.
6. Show that $\overrightarrow{SP} \cdot \overrightarrow{SQ} = 0$ that means the angle $\angle(PSQ) = 90^\circ$.
• Nov 30th 2011, 12:49 AM
eskimogenius
Re: Locus and the Parabola Problem
Having immense troubles here trying to determine the point of intersection of the parabola and the normal.

When you substitute it in, you have on one side, x^2, whilst on the other side, x. It is impossible to make x the subject of the equation.

How do I complete your step 5?

In other attempts, I have tried to use a separate variable for Q, with P being (2ap, ap^2) and Q being (2aq, aq^2). And then using Pythagoras' Theorem to prove true for the right angle. However, the algebra becomes intense and you are unable to eliminate and cancel down to the RHS. Also, I have attempted to use the gradients, then attempting to substitute the given condition of QS = 2PS, however, that fails also with the algebra coming up to powers of 4.
• Nov 30th 2011, 02:09 AM
earboth
Re: Locus and the Parabola Problem
Quote:

Originally Posted by eskimogenius
Having immense troubles here trying to determine the point of intersection of the parabola and the normal.

When you substitute it in, you have on one side, x^2, whilst on the other side, x. It is impossible to make x the subject of the equation.

How do I complete your step 5?

...

The intersection of the parabola and the normal:

equation of parabola: $y = \frac14 x^2$

equation of normal: $y = -\frac2p x+2+\frac14p^2$

Solve for x:

$\frac14 x^2 = -\frac2p x + 2 + \frac14p^2$

$x^2+\frac8p x - 8 - p^2=0$

$x = \frac{-\frac8p \pm \sqrt{\frac{64}{p^2} - 4(-8-p^2)}}2$

$x = \frac{-\frac8p \pm \frac2p \cdot \sqrt{p^4+8p^2+16 }}2$

The radicand is a complete square!

Therefore you'll get:

$\underbrace{x = p}_{P} ~\lor ~ \underbrace{x = -p-\frac8p}_{Q}$

So Q has the coordinates $Q\left(-p-\frac8p, \frac14\left(p+\frac8p \right)^2\right)$

To prove the orthogonality of $\overline{SP}$ and $\overline{SQ}$ respectively I would use vectors as I've described in my previous post.