Exercise 5.2 Class 12 Math Solutions - Unit 5 Analytical Geometry, Punjab Board
Step-by-step solutions to Exercise 5.2, Unit 5: Analytical Geometry, for Class 12 Punjab Board students. Covers calculating the angle between two intersecting lines, parallel and perpendicular conditions, and determining the area and collinearity of triangular regions from vertex coordinates, worked out question by question.
Exercise Overview
Exercise 5.2 of Class 12 Mathematics focuses on angular relationships between straight lines and the computation of planar triangular areas under the Punjab Board PCTB / PECTAA curriculum. Given two non-vertical lines with slopes (m_1) and (m_2), the acute angle (\theta) between them is derived geometrically as (\tan\theta = \left|\frac{m_2 - m_1}{1 + m_1 m_2}\right|). This formula directly yields the analytical criteria for Parallel Lines ((m_1 = m_2)) and Perpendicular Lines ((m_1 m_2 = -1) or (1 + m_1 m_2 = 0)).
The second part of the exercise investigates the Area of a Triangular Region bounded by vertices (A(x_1, y_1)), (B(x_2, y_2)), and (C(x_3, y_3)). Students utilize the elegant 3x3 determinant formula (\text{Area}(\Delta) = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|). This determinant formulation provides a definitive algebraic test for Collinearity: three distinct points lie on the exact same straight line if and only if the area of the triangle formed by them is zero.
For Punjab Board HSSC Part 2 examinations across all nine educational divisions (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 5.2 frequently provides essential Section B Short Questions and Long Question sub-parts. These step-by-step solutions provide clean slope conversions, explicit tangent formula substitutions, and complete determinant evaluations for every problem.