# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- # **Key Concepts** This exercise focuses on the following concepts: - Joint equations representing pairs of straight lines through the origin - Homogeneous equations of second degree: $ax^2 + 2hxy + by^2 = 0$ - Decomposing joint equations into separate line equations - Angle between pair of lines - Nature of lines: coincident ($h^2 = ab$), distinct real ($h^2 > ab$), and imaginary ($h^2 < ab$) - Condition for perpendicular lines ($a + b = 0$) - Construction of joint equations for perpendicular lines --- # **Important Formulas** Below are the key formulas used in this exercise: **General Form of Joint Equation:** $$ax^2 + 2hxy + by^2 = 0$$ **Angle Between Lines:** $$\tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|$$ **Conditions for Nature of Lines:** | Condition | Nature of Lines | | :--- | :--- | | $h^2 > ab$ | Two distinct real lines | | $h^2 = ab$ | Coincident lines | | $h^2 < ab$ | Imaginary lines (no real slope) | **Perpendicularity Condition:** $$a + b = 0$$ **Joint Equation of Perpendicular Lines:** If $ax^2 + 2hxy + by^2 = 0$ represents a pair of lines, the joint equation of lines through origin perpendicular to them is: $$bx^2 - 2hxy + ay^2 = 0$$ --- # **Summary** This exercise explores the analysis of joint equations representing pairs of straight lines passing through the origin. The key strategy involves recognizing the homogeneous second-degree form $ax^2 + 2hxy + by^2 = 0$ and applying the discriminant $h^2 - ab$ to determine the nature of the lines. For finding separate equations, factorize or use the quadratic in $m = y/x$. To find angles, apply the formula involving $h^2 - ab$ and $a+b$. Remember that perpendicularity is simply checked by $a+b=0$. When constructing joint equations for perpendicular lines, interchange the coefficients of $x^2$ and $y^2$ and reverse the sign of the $xy$ term.