4.5 Q-8

Question Statement

We are tasked with finding the equation of lines through the origin that are perpendicular to a given conic equation. The equation provided is:

$$a x^{2} + 2 h x y + b y^{2} = 0$$

We need to derive the equations of lines that pass through the origin and are perpendicular to the line described by the given equation.

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Background and Explanation

To solve this problem, we will apply principles from conic sections and the concept of perpendicular lines. A conic equation of the form $ax^2 + 2hxy + by^2 = 0$ describes a general quadratic curve. To find lines perpendicular to a line defined by such an equation, we first need to transform the given quadratic equation into a form where the slope of the line can be easily determined.

We will use the standard method of solving quadratic equations and work with the slope form of the lines derived from the equation of the conic.

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Solution

Step 1: Rearrange the equation to express it in terms of $\frac{y}{x}$

Starting from the given equation:

$$a x^{2} + 2 h x y + b y^{2} = 0$$

We divide the entire equation by $b x^2$:

$$\frac{b y^{2}}{b x^{2}} - \frac{2 h x y}{b x^{2}} + \frac{a x^{2}}{b x^{2}} = 0$$

This simplifies to:

$$\left[\frac{y}{x}\right]^{2} + \frac{2 h}{b} \left[\frac{y}{x}\right] + \frac{a}{b} = 0$$

This is a quadratic equation in $\frac{y}{x}$.

Step 2: Solve the quadratic equation

We solve for $\frac{y}{x}$ using the quadratic formula:

$$\frac{y}{x} = \frac{-\frac{2h}{b} \pm \sqrt{\left(\frac{2h}{b}\right)^2 - 4 \cdot 1 \cdot \frac{a}{b}}}{2 \cdot 1}$$

Simplifying the discriminant:

$$\frac{y}{x} = \frac{-\frac{2h}{b} \pm \sqrt{\frac{4h^2}{b^2} - \frac{4a}{b}}}{2}$$

Factor out the common terms:

$$\frac{y}{x} = \frac{2h}{b} \pm \sqrt{\frac{h^2 - ab}{b^2}}$$

Simplifying further:

$$\frac{y}{x} = \frac{-h \pm 2 \sqrt{h^2 - ab}}{b}$$

This gives two possible slopes for the perpendicular lines:

1. $m_1 = \frac{-h - \sqrt{h^2 - ab}}{b}$
2. $m_2 = \frac{-h + \sqrt{h^2 - ab}}{b}$

Step 3: Find the slope of the required perpendicular line

For the line perpendicular to the given conic, the slope is:

$$m_3 = \frac{b}{-h + \sqrt{h^2 - ab}}$$

This is the slope of the line through the origin that is perpendicular to the given conic equation.

Step 4: Derive the equation of the perpendicular line

The equation of the line with slope $m_3$ through the origin is:

$$y = m_3 x$$

Substituting the value of $m_3$:

$$y = \frac{b}{-h + \sqrt{h^2 - ab}} x$$

This is the equation of one of the perpendicular lines.

Step 5: Repeat for the second line

Similarly, for the second perpendicular line, we use the other slope:

$$y = \frac{-h - \sqrt{h^2 - ab}}{b} x$$

Thus, the second equation of the perpendicular line is:

$$\left( h - \sqrt{h^2 - ab} \right) y = b x$$

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Key Formulas or Methods Used

• Quadratic Formula: Used to solve the quadratic equation in terms of $\frac{y}{x}$.
• Perpendicular Line Formula: The equation for a line through the origin is $y = m x$, where $m$ is the slope.

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Summary of Steps

1. Start with the given equation of the conic: $a x^2 + 2 h x y + b y^2 = 0$.
2. Rearrange the equation and express it in terms of $\frac{y}{x}$.
3. Solve the resulting quadratic equation for $\frac{y}{x}$.
4. Calculate the slopes $m_1$ and $m_2$ of the lines defined by the quadratic equation.
5. Find the slope of the required perpendicular line $m_3$.
6. Derive the equation of the perpendicular lines through the origin using the slope $m_3$.

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