Question Statement

Show that the lines represented by the following equations are coincident or distinct or imaginary.

(i) $x^2+4xy-21y^2=0$

(ii) $4x^2-12xy+9y^2=0$

(iii) $x^2+xy+y^2=0$

(iv) $x^2-9y^2=0$

---

Background and Explanation

A general homogeneous equation of second degree $a{x}^2+2hxy+b{y}^2=0$ represents a pair of straight lines passing through the origin. The nature of these lines—whether they are real and distinct, coincident, or imaginary—is determined by the discriminant $h^2-ab$.

---

Solution

Part (i): $x^2+4xy-21y^2=0$

Compare the given equation with the standard form $a{x}^2+2hxy+b{y}^2=0$:

$a=1,b=-21,2h=4⟹h=2$

Now calculate the discriminant:

$h^2-ab=(2{)}^2-(1)(-21)=4+21=25$

Since $h^2-ab=25>0$, the discriminant is positive. Therefore, the lines represented by this equation are real and distinct.

Part (ii): $4x^2-12xy+9y^2=0$

Compare the given equation with the standard form $a{x}^2+2hxy+b{y}^2=0$:

$a=4,b=9,2h=-12⟹h=-6$

Now calculate the discriminant:

$h^2-ab=(-6{)}^2-(4)(9)=36-36=0$

Since $h^2-ab=0$, the discriminant equals zero. This indicates that the two lines merge into one, so the lines are real and coincident.

Part (iii): $x^2+xy+y^2=0$

Compare the given equation with the standard form $a{x}^2+2hxy+b{y}^2=0$:

$a=1,b=1,2h=1⟹h=\frac{1}{2}$

Now calculate the discriminant:

$h^2-ab={\left(\frac{1}{2}\right)}^2-(1)(1)$

$\begin{array}{rl}& =\frac{1}{4}-1\\ & =-\frac{3}{4}\end{array}$

Since $h^2-ab=-\frac{3}{4}<0$, the discriminant is negative. Therefore, the lines are imaginary (they have no real existence, though they may be considered as conjugate complex lines).

Part (iv): $x^2-9y^2=0$

Compare the given equation with the standard form $a{x}^2+2hxy+b{y}^2=0$:

$a=1,b=-9,2h=0⟹h=0$

Now calculate the discriminant:

$h^2-ab=(0{)}^2-(1)(-9)=0+9=9$

Since $h^2-ab=9>0$, the discriminant is positive. Therefore, the lines are real and distinct (physically, this represents the pair of lines $x=\pm 3y$).

---

Key Formulas or Methods Used

• Standard form of homogeneous equation of second degree: $a{x}^2+2hxy+b{y}^2=0$
• Discriminant condition: $\Delta =h^2-ab$
• Nature of lines classification:
  • If $h^2-ab>0$: Real and distinct lines
  • If $h^2-ab=0$: Real and coincident lines
  • If $h^2-ab<0$: Imaginary lines

---

Summary of Steps

1. Identify coefficients: Compare the given equation with $a{x}^2+2hxy+b{y}^2=0$ to determine the values of $a$, $b$, and $h$ (remembering that the coefficient of $xy$ is $2h$, not $h$).
2. Calculate discriminant: Compute $h^2-ab$ using the identified coefficients.
3. Interpret the result:
   • Positive value $\to$ Real and distinct lines
   • Zero $\to$ Real and coincident lines
   • Negative value $\to$ Imaginary lines