05_Ex 4.5

Exercise Questions

Questions	Links
Q1. $10 x^{2}-23 x y-5 y^{2}=0$	4.5 Q-1
Q2. $3 x^{2}+7 x y+2 y^{2}=0$	4.5 Q-2
Q3. $9 x^{2}+24 x y+16 y^{2}=0$	4.5 Q-3
Q4. $2 x^{2}+3 x y-5 y^{2}=0$	4.5 Q-4
Q5. $6 x^{2}-19 x y+15 y^{2}=0$	4.5 Q-5
Q6. $\mathrm{x}^{2}+2 x y \sec x+\mathrm{y}^{2}=0$	4.5 Q-6
Q7. Find a joint equation of the lines…	4.5 Q-7
Q8. Find an equation of the lines through…	4.5 Q-8
Q9. Find the area of the region bounded by…	4.5 Q-9

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Overview

This Exercise focuses on solving homogeneous quadratic equations, finding the slopes and angles between lines, and understanding the geometric implications of second-degree equations. These concepts are critical for analyzing linear systems, intersections, and angles in two-dimensional geometry.

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Key Concepts

1. Homogeneous Equations:
   • A quadratic equation of the form:

$$ax^2 + 2hxy + by^2 = 0$$

 represents a pair of straight lines passing through the origin.

• The degree of the equation is determined by the sum of powers in each term.

2. Factoring Quadratic Equations:
   • These can often be expressed as:

$$(px + qy)(rx + sy) = 0$$

 aiding in finding individual lines by solving each factor.

3. Slope of Lines:

• The slope $m$ is defined as:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• For perpendicular lines, the product of slopes equals $-1$.

4. Trigonometric Relationship for Angles:
   • The tangent of the angle $\theta$ between two lines is given by:

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

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Important Formulas

1. Slope Relationships:
   For the equation $ax^2 + 2hxy + by^2 = 0$, the slopes $m_1$ and $m_2$ of the lines satisfy:

$$m_1 + m_2 = -\frac{2h}{a}, \quad m_1m_2 = \frac{b}{a}.$$

2. Quadratic Formula:
   For $ax^2 + bx + c = 0$, roots are given by:

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

3. Angle Between Lines:
   For two intersecting lines, $\theta$ can be calculated as:

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

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Tips and Tricks

• Identifying Homogeneous Equations:
  Ensure all terms are of the same degree.
• Factoring Quadratics:
  Always simplify and factorize to find explicit line equations.
• Angles Between Lines:
  Use $\tan \theta$ formula with care; ensure $a + b \neq 0$.
• Perpendicular Lines:
  Check the product of slopes for verification.

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Summary

This Exercise emphasizes solving quadratic equations representing lines, finding slopes, and calculating angles between lines. Key formulas such as $\tan \theta$ and the quadratic solution are essential tools. Practice factoring, analyzing slopes, and understanding perpendicularity for mastery.

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Reference

By Sir Shahzad Sair:

By Great Science Academy:

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