4.5 Q-1

Question Statement

Given the equation of a pair of lines:
$10x^2 - 23xy - 5y^2 = 0,$

1. Rewrite it as two linear equations.
2. Find the measure of the angle between the lines.

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Background and Explanation

The equation represents two straight lines passing through the origin. To find the individual lines, we express the equation in terms of the slope $\frac{y}{x}$. Once the slopes are determined, we use the formula for the angle between two lines, $\tan \theta = \frac{|m_1 - m_2|}{1 + m_1 m_2}$ or its equivalent form for general equations of lines.

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Solution

Step 1: Rewrite the equation in terms of $\frac{y}{x}$

Divide the entire equation by $x^2$ (assuming $x \neq 0$):

$$10 + 23\left(\frac{y}{x}\right) + 5\left(\frac{y}{x}\right)^2 = 0.$$

Let $\frac{y}{x} = m$. The equation becomes a quadratic equation in $m$:

$$5m^2 + 23m - 10 = 0.$$

Step 2: Solve for $m$ (slopes)

Using the quadratic formula $m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

$$m = \frac{-23 \pm \sqrt{23^2 - 4(5)(-10)}}{2(5)}.$$

Substitute values:

$$m = \frac{-23 \pm \sqrt{529 + 200}}{10},$$ $$m = \frac{-23 \pm \sqrt{729}}{10},$$ $$m = \frac{-23 \pm 27}{10}.$$

Solve for $m$:

$$m_1 = \frac{-23 + 27}{10} = \frac{4}{10} = \frac{2}{5}, \quad m_2 = \frac{-23 - 27}{10} = \frac{-50}{10} = -5.$$

Step 3: Write the equations of the lines

Using $y = mx$, the equations of the lines are:

$$y = \frac{2}{5}x \quad \text{or} \quad 2x - 5y = 0,$$ $$y = -5x \quad \text{or} \quad 5x + y = 0.$$

Step 4: Find the angle between the lines

The general equation of a pair of lines is given by $ax^2 + 2hxy + by^2 = 0$. Here, $a = 10$, $2h = -23$ (so $h = -\frac{23}{2}$), and $b = -5$.
The angle between the lines can be found using:

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

Substitute the values:

$$\tan \theta = \frac{2\sqrt{\left(-\frac{23}{2}\right)^2 - (10)(-5)}}{10 + (-5)}.$$

Simplify:

$$\tan \theta = \frac{2\sqrt{\frac{529}{4} + 50}}{5}.$$ $$\tan \theta = \frac{2\sqrt{\frac{729}{4}}}{5}.$$ $$\tan \theta = \frac{2 \cdot \frac{27}{2}}{5}.$$ $$\tan \theta = \frac{27}{5}.$$

Finally, calculate $\theta$:

$$\theta = \tan^{-1}\left(\frac{27}{5}\right) \approx 79.51^\circ.$$

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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 (general form):
   $\tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b}.$

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Summary of Steps

1. Rewrite the given equation as a quadratic equation in $m = \frac{y}{x}$.
2. Solve the quadratic equation for $m$ to find the slopes of the two lines.
3. Write the equations of the lines using their slopes.
4. Use the formula for $\tan \theta$ to calculate the angle between the lines.
5. Approximate the angle using the inverse tangent function.