How to Find the Equation of Straight Lines Through the Origin Perpendicular to Given Lines

When dealing with analytical geometry, one often needs to work with the equations of lines, especially when they involve quadratic expressions. This article will guide you through the process of finding the equation of lines that pass through the origin and are perpendicular to the lines represented by a given quadratic equation.

Introduction

The given quadratic equation is:

(x^2 - 5xy - 4y^2 0)

The steps we will follow to solve this problem are:

Factor the quadratic equation. Find the slopes of the lines represented by the quadratic equation. Find the slopes of the lines that are perpendicular to those lines. Write the equations of the perpendicular lines through the origin. Combine these equations into a single equation.

Step 1: Factor the Quadratic Equation

The given quadratic equation is:

(x^2 - 5xy - 4y^2 0)

This equation can be factored into two linear factors. We can rewrite it as:

((x - ay)(x - by) 0)

where (a) and (b) are the slopes of the lines. To find (a) and (b), we can use the quadratic formula:

(t^2 - 5t - 4 0)

where (t frac{y}{x}) is the slope of the lines. Using the quadratic formula:

(t frac{-(-5) pm sqrt{(-5)^2 - 4 cdot 1 cdot (-4)}}{2 cdot 1} frac{5 pm sqrt{25 16}}{2} frac{5 pm sqrt{41}}{2})

This gives us:

(t_1 frac{5 sqrt{41}}{2}) and (t_2 frac{5 - sqrt{41}}{2})

Therefore, the slopes of the lines represented by the equation are:

(m_1 frac{5 sqrt{41}}{2}) and (m_2 frac{5 - sqrt{41}}{2})

Step 2: Find the Slopes of the Perpendicular Lines

The slopes of the lines that are perpendicular to these lines can be found using the negative reciprocal. Therefore, the slopes of the perpendicular lines are:

(m_{p1} -frac{2}{5 sqrt{41}}) and (m_{p2} -frac{2}{5 - sqrt{41}})

Step 3: Write the Equations of the Perpendicular Lines Through the Origin

The equations of the lines through the origin with these slopes can be expressed as:

(y m_{p1}x) and (y m_{p2}x)

Substituting the slopes, we get:

(y -frac{2}{5 sqrt{41}}x) and (y -frac{2}{5 - sqrt{41}}x)

Step 4: Combine into a Single Equation

The single equation that represents both lines can be written in the form:

(y frac{2}{5 sqrt{41}}x 0) and (y frac{2}{5 - sqrt{41}}x 0)

To combine these, we can use the fact that both can be expressed as:

(5y (2/5 sqrt{41})x 0) and (5y (2/5 - sqrt{41})x 0)

Thus, the single equation of the lines through the origin that are perpendicular to the original lines is:

(y frac{1}{4}x 0) or (4y - x 0)

This gives you the two equations of the lines through the origin that are perpendicular to the original lines represented by the quadratic equation (x^2 - 5xy - 4y^2 0).

Conclusion

By using the steps outlined above, we can effectively find the equations of the lines through the origin that are perpendicular to a given quadratic equation. This process is particularly useful in the field of analytical geometry and can help solve a wide range of problems involving straight lines.

If you need further assistance or have more questions on this topic, feel free to explore more resources or consult with a math expert.