Straight lines

[Abdul Quadeer]

1. The problem statement, all variables and given/known data

If the pair of lines x^2-2pxy-y^2=0 and x^2-2qxy-y^2=0 are such that each pair bisects the angle between the other pair, then what is the value of pq?

2. Relevant equations



3. The attempt at a solution

I don't understand what 'each pair bisects the angle between the other pair' mean.
Is it that each line of first pair bisects the angle between the two lines of other pair (obtuse and acute angle)?.
How do I proceed with this question?

[Mentallic]

Yes pretty much. The lines in each equation are perpendicular to each other and this can be proven by showing that the first line satisfies:

y=(-p\pm\sqrt{p^2+1})x and thus if m_1=-p+\sqrt{p^2+1} and m_2=-p-\sqrt{p^2+1} then m_1m_2=-1

And obviously since it's saying that p and q are chosen such that each pair of lines bisect each other, if we take any one line, then the other line from the other equation will have an angle difference of \pi/4 radians (since bisecting an angle of \pi/2 is half of that).

So we have m_p=-p\pm\sqrt{p^2+1} and m_q=-q\pm\sqrt{q^2+1}.

since the gradient m_p=tan\theta then m_q=tan(\theta+\frac{\pi}{4})=\frac{1+tan\theta}{1-tan\theta}=\frac{1+m_p}{1-m_p}

Can you proceed from here?

[Abdul Quadeer]

In my book, it is done this way.

The second pair must be identical with (x^2-y^2)/[1-(-1)] = xy/-p

i.e. x^2 + (2/p)xy - y^2=0. Consequently, 2/p=-2q
i.e. pq=-1

I don't understand any thing from this. What is done here?
And what is the next step in your method?

[Mentallic]

Neither can I :bugeye:

"The second pair must be identical with (x^2-y^2)/[1-(-1)] = xy/-p"

This line has completely stumped me on what they mean. Does it give any other information?

For my solution we have the equation

m_q=\frac{1+m_p}{1-m_p} where m_q=-q\pm\sqrt{q^2+1}, m_p=-p\pm\sqrt{p^2+1}

Substituting mq into the equation:

-q\pm\sqrt{q^2+1}=\frac{1+m_p}{1-m_p}

Now solve for q, simplify, then substitute for mp and simplify further till you get the desired result.

[Abdul Quadeer]

Unfortunately, nothing is mentioned.
Anyway I got the answer by your method (though it is a bit long :p)
Thanks alot.

[Mentallic]

I agree, it is. I'd also like to know what your book meant by that... You gotta love how they don't even bother to give a reasonable explanation, while with other solutions they take you through it step by step so slowly, it makes you jump pages at a time to get to the point.

[Abdul Quadeer]

Yeah, very true.

[Abdul Quadeer]

You wanted to know what my book meant by that...
For ax^2 +2hxy + by^2 = 0 the angle bisector is given as...

x^2 - y^2 / ( a-b) = xy / h

This is a formula I got from my friend.

[Mentallic]

And how was that formula derived?