4.5 Q-3

Question Statement

Solve the given quadratic equation of the form: $9x^2 + 24xy + 16y^2 = 0.$

Determine the slopes of the lines it represents and the measure of the angle between the lines.

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Background and Explanation

This is a homogeneous quadratic equation in $x$ and $y$, which represents two straight lines passing through the origin. To solve such equations, we assume a ratio $\frac{y}{x} = m$ (where $m$ is the slope) and substitute it into the equation. This reduces the equation to a quadratic in $m$, which we can solve to find the slopes of the lines.

If the two slopes are equal, the lines are parallel, and the angle between them is zero.

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Solution

Step 1: Rewrite the Equation in Terms of $\frac{y}{x}$

Substitute $\frac{y}{x} = m$ into the equation $9x^2 + 24xy + 16y^2 = 0$:

$$16m^2 + 24m + 9 = 0.$$

This is now a quadratic equation in $m$.

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Step 2: Solve for $m$ Using the Quadratic Formula

The quadratic formula is:

$$m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$

Here, $a = 16$, $b = 24$, and $c = 9$. Substitute these values:

$$m = \frac{-(24) \pm \sqrt{(24)^2 - 4(16)(9)}}{2(16)}.$$

Simplify:

$$m = \frac{-24 \pm \sqrt{576 - 576}}{32}.$$ $$m = \frac{-24 \pm 0}{32}.$$ $$m = -\frac{3}{4}.$$

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Step 3: Interpret the Results

The solution $m = -\frac{3}{4}$ appears twice, meaning the quadratic has a double root. This indicates the equation represents two lines with the same slope:

$$y = -\frac{3}{4}x \quad \text{or} \quad 3x + 4y = 0.$$

Thus, the lines are identical and overlap.

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Step 4: Measure the Angle Between the Lines

When two lines are parallel or identical, the angle between them is zero. Hence, the measure of the acute angle is:

$$\theta = 0^\circ.$$

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Key Formulas or Methods Used

1. Quadratic Substitution: $\frac{y}{x} = m$ simplifies the equation.
2. Quadratic Formula: $m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$
3. Angle Between Lines: Parallel lines have an angle of $0^\circ$.

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Summary of Steps

1. Substitute $\frac{y}{x} = m$ into the given equation.
2. Solve the resulting quadratic equation for $m$.
3. Identify the slopes of the lines.
4. Conclude that the lines are parallel (or identical) since they share the same slope.
5. State that the angle between the lines is $0^\circ$.