Finding the Value of pq When Lines Bisect Each Other

• zorro
In summary: I don't understand what 'angle bisector' means. What is the next step in your method?"For ax^2 +2hxy + by^2 = 0 the angle bisector is given as...x^2 - y^2 / ( a-b) = xy / h"This line is very similar to the one in the homework statement. It states that the angle bisector is the line that divides the angle in two equal parts.
zorro

Homework Statement

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?

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?

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 anyone 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?

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?

Neither can I

"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.

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

I agree, it is. I'd also like to know what your book meant by that... You got to 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.

Yeah, very true.

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.

And how was that formula derived?

1. What is the definition of "Finding the Value of pq When Lines Bisect Each Other"?

"Finding the Value of pq When Lines Bisect Each Other" is a geometric problem that involves finding the value of the line segment pq when two lines bisect each other at point p and q.

2. How do you solve for the value of pq in this problem?

The value of pq can be solved by using the properties of bisectors, such as the angle bisector theorem and the segment bisector theorem. These theorems state that when a line bisects an angle or a line segment, it divides them into two equal parts.

3. What are some common strategies for solving this problem?

Some common strategies for solving "Finding the Value of pq When Lines Bisect Each Other" include using algebraic equations, setting up proportions, and using the Pythagorean theorem. These methods can help to find the value of pq by relating it to other known values in the problem.

4. What are the key steps to solving this problem?

The key steps to solving "Finding the Value of pq When Lines Bisect Each Other" are identifying the bisected angles or line segments, setting up the appropriate equations or proportions, and solving for the value of pq. It is also important to check the solution and make sure it satisfies all the given conditions in the problem.

5. Why is "Finding the Value of pq When Lines Bisect Each Other" an important concept in geometry?

This concept is important in geometry because it helps us understand the relationships between angles and line segments when bisected. It also allows us to solve more complex geometric problems by using the properties of bisectors. Additionally, this concept is useful in real-world applications such as architecture and engineering.

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