Usually, in any subject, we always begin with simpler things and then move to study more complex items. The same applies to the study of a pair of straight lines.
Here, we will study a pair of lines that pass through the origin. Such a pair of lines results in a simpler form of various results. Specifically, the combined equation is a homogeneous equation of degree two. We will start with the basic concept of the homogeneous equation.
What is a Homogeneous Equation?
To understand this concept, first, we have to learn about the degree of a term.
Degree of a Term
The degree of a term is the sum of indices (powers) of all variables in the term.
For example, in the equation $2x^2 + 5 x^2y+3y+7$, we have the following
- Four terms $2x^2, 5 x^2y, 3y$ and 7.
- The degree of the term $2x^2$ is 2.
- The degree of the term $5 x^2y$ is 3.
- The degree of the term $3y$ is 1.
- The degree of the term $7$ is 0.
It is to note that the degree of $0$ is undefined.
Homogeneous Equation
If all terms of an equation have the same degree, then the equation is known as a homogeneous equation.
For example, $x^2 + 2 xy + 3 y^2 = 0$ is a homogeneous equation while $x + y – 2 = 0$ is not.
Homogeneous Equation of Degree Two
A homogeneous equation of degree two is an equation where all terms have degree two. As an example, the combined equation of a pair of lines that pass through the origin is a homogeneous equation of degree two. Let’s see a proof of it.
An equation of a line that passes through the origin is of form $ax+by = 0$ (why?). The constant term in the equation is zero, and the origin $(0,0)$ satisfies the equation.
Let $a_1 x + b_1y = 0$ and $a_2 x + b_2 y = 0$ be a pair of lines through the origin. The combined equation of this pair of lines is given by
\begin{aligned}
(a_1 x + b_1y) (a_2 x + b_2 y) &= 0, \\
a_1 a_2 x^2 + a_1 b_2 xy + a_2 b_1 xy + b_1 b_2 y^2 &= 0,\\
a_1 a_2 x^2 + (a_1 b_2 + a_2 b_1 )xy + b_1 b_2 y^2 &= 0.
\end{aligned}
One can easily verify that the above equation is a homogeneous equation of degree two.
Notation
Generally, we denote a homogeneous equation of degree two in a standard fashion as
\begin{aligned}
a x^2 + 2hxy + b y^2 = 0.
\end{aligned}
Condition for Representing a Pair of Lines
As we have seen, the combined equation of a pair of lines that pass through the origin is always a homogeneous equation of degree two. However, the converse is not always true. Every homogeneous equation of degree two need not represent a pair of lines.
For example, $x^2 + y^2 = 0$ does not represent a pair of lines as it is an equation of a circle with radius zero.
We present the condition for representing a pair of lines as a homogeneous equation of degree two in the form of a theorem.
Theorem
A homogeneous equation of degree two given by $ax^2 + 2hxy+by^2 = 0$ represents a pair of lines through origin if $h^2 – ab \geq 0$.
Proof: If the equation represents a pair of lines through origin, then $y = mx$ should satisfy it where $m$ denotes slope. Substituting this, we get
\begin{aligned}
b m^2 x^2 + 2 h m x^2 + a x^2&= 0, \\
x^2 (b m^2 + 2hm + a) &=0.
\end{aligned}
The above equation should get satisfied for all values of $x$. This implies
\begin{aligned}
b m^2 + 2hm + a &= 0.
\end{aligned}
Assuming $b \neq 0$, we can observe that the above equation is a quadratic equation whose solutions are given by
\begin{aligned}
m &= \frac{-2h \pm \sqrt{4h^2 – 4ab}}{2b}, \\
&= \frac{-h \pm \sqrt{h^2 – ab}}{b}.
\end{aligned}
The solutions provide us the slopes of the pair of lines. We get a solution only when $h^2 – ab \geq 0$; otherwise, the slopes would turn out to be complex numbers. It is to note that we deal with real numbers only in coordinate geometry.
If $b = 0$, then the condition $h^2 – ab \geq 0$ still remains valid and is always true. So, we must prove that the homogeneous equation always represents a pair of lines with $b = 0$.
With $ b = 0$, the homogeneous equation can be written as
If $ b = 0$, then the homogeneous equation becomes
\begin{aligned}
a x^2 + 2 hxy &= 0, \\
x(ax+2hy) &=0.
\end{aligned}
Thus, it represents a pair of lines $x = 0$ and $ax + 2hy = 0$ with $b = 0$.
Auxiliary Equation
We arrived at an equation $b m^2 + 2 hm + a = 0$ while proving the condition for representing a pair of lines. This equation is known as an auxiliary equation. As the word ‘auxiliary’ means supplementary or additional help, this equation is of tremendous help to us in solving problems related to pairs of straight lines.
A few important results that follow from the auxiliary equation are:
1. If $h^2 – ab \geq 0$ and $ b \neq 0$, then two lines will have slopes
\begin{aligned}
m_1 = \frac{-h + \sqrt{h^2 – ab}}{b}, \quad m_2 = \frac{-h – \sqrt{h^2 – ab}}{b}.
\end{aligned}
The sum and product of the slopes are
\begin{aligned}
m_1 + m_2 = \frac{-2h}{b}, \quad m_1 m_2 &= \frac{a}{b}.
\end{aligned}
2. If $b = 0$, then two lines will have slopes
\begin{aligned}
m_1 = \infty (\text{undefined}), \quad m_2 = \frac{-a}{2h}, h \neq 0.
\end{aligned}
The first line is $x = 0$ and second line is $ y = \frac{-a}{2h} x$.
3. If $h^2 – ab = 0$, then two lines will coincide, i.e., they will be identical.
4. If $h^2 – ab = 0$, then two lines will be distinct.
Examples
Ex. 1: Find separate equations of lines represented by $x^2 + 2xy – y^2 = 0$.
Solution: Here $a = 1, h = 1$ and $b = -1$. The equation satisfies the condition $h^2 – ab \geq 0$ to be a pair of lines. The slopes of the lines would be
\begin{aligned}
m_1 = 1 + \sqrt{2}, \quad m_2 = 1-\sqrt{2}.
\end{aligned}
Thus, the separate equations would be $(1+\sqrt{2})x – y = 0$ and $(1-\sqrt{2})x – y = 0$.
Ex. 2: Find the value of $k$ if slops of lines represented by $3 x^2 + k xy-y^2 = 0$ differ by 4.
Solution: We have $a = 3$, $h = \frac{k}{2}$ and $b = -1$. It is easy to verify that $h^2 – ab \geq 0$ always irrespective of value of $k$. Here we are given that $m_1 – m_2 = \pm 4$. So, using a relation
\begin{aligned}
(m_1 + m_2)^2 = (m_1 – m_2)^2 + 4 m_1 m_2.
\end{aligned}
We have $m_1 + m_2 = \frac{-2h}{b} = k$ and $m_1 m_2 = -3$. Using these, we get
\begin{aligned}
k^2 &= 16 – 4 \times 3, \\
&= 4.
\end{aligned}
Thus, $k = \pm 2$ are two possible values.
Summary
In this article, we learned first about what is a homogeneous equation. Then, we looked at a specific homogeneous equation of degree two, which represents a pair of straight lines. We also learned about the auxiliary condition, which is of great help in the derivation of various results.
If you have any questions, you can drop a comment below. I will be delighted to answer your queries! Have happy learning!