4.5 Q-2

Question Statement

Solve the quadratic equation $3x^2 + 7xy + 2y^2 = 0$ to determine the lines it represents and the angle between them.

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Background and Explanation

This question involves solving a homogeneous quadratic equation in $x$ and $y$. To determine the lines it represents, we rewrite the equation in terms of $\frac{y}{x} = m$, which corresponds to the slopes of the lines. We then use the formula for the angle between two lines:
$\tan\theta = \frac{|m_1 - m_2|}{1 + m_1 m_2},$
or equivalently for a homogeneous equation:
$\tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b},$
where $a$, $h$, and $b$ are coefficients of $x^2$, $xy$, and $y^2$, respectively.

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Solution

Step 1: Rewrite the equation

Rewrite the quadratic equation in terms of $\frac{y}{x} = m$:

$$2\left(\frac{y}{x}\right)^2 + 7\left(\frac{y}{x}\right) + 3 = 0$$

This becomes:

$$2m^2 + 7m + 3 = 0$$

Step 2: Solve for $m$

Using the quadratic formula, $m = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$:

$$m = \frac{-7 \pm \sqrt{7^2 - 4(2)(3)}}{2(2)}$$

Simplify step by step:

1. Calculate the discriminant:

$$\sqrt{49 - 24} = \sqrt{25} = 5$$

2. Substitute into the formula:

$$m = \frac{-7 \pm 5}{4}$$

3. Find the two roots:

$$m = \frac{-7 + 5}{4} = -\frac{2}{4} = -\frac{1}{2}$$ $$m = \frac{-7 - 5}{4} = -\frac{12}{4} = -3$$

Thus, $m_1 = -\frac{1}{2}$ and $m_2 = -3$.

Step 3: Find the equations of the lines

The lines are given by:

$$y = m_1x \quad \text{and} \quad y = m_2x$$

Substitute $m_1$ and $m_2$:

1. Line 1: $y = -\frac{1}{2}x \quad \Rightarrow \quad x + 2y = 0$
2. Line 2: $y = -3x \quad \Rightarrow \quad 3x + y = 0$

Step 4: Find the angle between the lines

The formula for the angle between two lines in a homogeneous quadratic equation is:

$$\tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b}$$

Substitute $a = 3$, $h = \frac{7}{2}$, and $b = 2$:

$$\tan\theta = \frac{2\sqrt{\left(\frac{7}{2}\right)^2 - (3)(2)}}{3 + 2}$$

Simplify step by step:

1. Square $h$:

$$\left(\frac{7}{2}\right)^2 = \frac{49}{4}$$

2. Subtract $ab$:

$$\frac{49}{4} - 6 = \frac{25}{4}$$

3. Take the square root:

$$\sqrt{\frac{25}{4}} = \frac{5}{2}$$

4. Multiply by 2 and divide by $a + b$:

$$\frac{2\left(\frac{5}{2}\right)}{5} = \frac{5}{5} = 1$$

Thus:

$$\tan\theta = 1 \quad \Rightarrow \quad \theta = \tan^{-1}(1) = 45^\circ$$

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Key Formulas or Methods Used

1. Quadratic Formula:
   $m = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$

2. Angle Between Two Lines in Homogeneous Form:
   $\tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b}$

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Summary of Steps

1. Rewrite the quadratic equation in terms of $m = \frac{y}{x}$.
2. Solve the quadratic equation for $m$ using the quadratic formula.
3. Write the equations of the lines using the roots of $m$.
4. Apply the formula for the angle between lines to calculate $\theta$.
5. Conclude that the angle between the lines is $45^\circ$.