Exercise 5.3 Class 12 Math Solutions - Unit 5 Analytical Geometry, Punjab Board

Step-by-step solutions to Exercise 5.3, Unit 5: Analytical Geometry, for Class 12 Punjab Board students. Covers homogeneous second-degree equations representing pairs of lines through the origin (ax² + 2hxy + by² = 0), calculating the acute angle between them, and real-world applications of analytical geometry, worked out question by question.

This page provides complete solved solutions for Exercise 5.3 of Unit 5: Analytical Geometry, Class 12 Mathematics, for Punjab Board (PCTB / PECTAA). Every question is solved step by step, analyzing homogeneous quadratic equations \(ax^2 + 2hxy + by^2 = 0\) representing pairs of straight lines through the origin, calculating the acute angle between lines using \(\tan\theta = \frac{2\sqrt{h^2 - ab}}{|a + b|}\), applying orthogonality (\(a + b = 0\)) and coincidence (\(h^2 = ab\)) criteria, and solving practical real-world geometry modeling problems.
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Exercise Overview

Exercise 5.3 concludes Unit 5: Analytical Geometry in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum by exploring joint equations representing pairs of straight lines. A general homogeneous equation of the second degree in two variables, (ax^2 + 2hxy + by^2 = 0), represents a pair of straight lines passing through the origin ((0, 0)). Students factor the quadratic equation into two distinct linear equations (y - m_1 x = 0) and (y - m_2 x = 0), establishing the sum and product of slopes: (m_1 + m_2 = -\frac{2h}{b}) and (m_1 m_2 = \frac{a}{b}).

The exercise establishes the acute angle formula between the line pair as (\tan\theta = \frac{2\sqrt{h^2 - ab}}{|a + b|}). This directly generates two vital geometric conditions: the lines are Perpendicular (Orthogonal) if (a + b = 0) (the coefficient of (x^2) plus the coefficient of (y^2) equals zero), and the lines are Coincident (Parallel) if (h^2 - ab = 0) ((h^2 = ab)). Additionally, students study real-world applications of analytical geometry in surveying triangulation, architectural trusses, navigation vector paths, and ray tracing.

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 5.3 provides essential Section C Long Questions and high-frequency MCQs. These step-by-step solutions provide rigorous parameter extractions, explicit angle derivations, and clean geometric proofs for every problem.

FAQs

Are these Exercise 5.3 solutions according to the Punjab Board syllabus?
Yes. These solutions follow the Punjab Curriculum and Textbook Board (PCTB / PECTAA) syllabus for Class 12 Mathematics, matching the pair of straight lines topics in Unit 5, Exercise 5.3 for the 2026-2027 session.
Do these solutions cover every question in Exercise 5.3?
Yes, all questions from Exercise 5.3 are included, worked through step by step covering homogeneous second-degree equations, acute angles between line pairs, orthogonality/coincidence tests, and real-life applied geometry.
Are the solutions step-by-step?
Yes. Each question details the algebraic procedure-identifying parameters a, h, and b from ax² + 2hxy + by² = 0, applying the angle tangent formula tan θ = 2√(h² - ab) / |a + b|, testing perpendicularity a + b = 0, and formulating geometric models.
Can students read these solutions online?
Yes, the full Exercise 5.3 solution set is available to read online on this page without any download required.
Are these solutions based on the latest PCTB / PECTAA syllabus?
Yes, they are aligned with the current PCTB and PECTAA Mathematics syllabus for the 2026-2027 academic session for Class 12.
Are these solutions free for students?
Yes, Exercise 5.3 solutions on this page are completely free to read for all Class 12 students preparing for Lahore Board, Rawalpindi Board, Gujranwala Board, or any other Punjab Board examinations.