Exercise 6.4 Class 12 Math Solutions - Unit 6 Conics, Punjab Board

Step-by-step solutions to Exercise 6.4, Unit 6: Conics, for Class 12 Punjab Board students. Covers the definition of an ellipse with eccentricity e < 1, derivation of standard equation x²/a² + y²/b² = 1, key features (foci (±c, 0), major and minor axes, vertices, directrices x = ±a/e, latus rectum 2b²/a), translated horizontal and vertical forms, and equations of tangents and normals including the tangency condition c² = a²m² + b², worked out question by question.

This page provides complete step-by-step solved solutions for Exercise 6.4 of Unit 6: Conics, Class 12 Mathematics, for Punjab Board (PCTB / PECTAA). All textbook problems are solved with complete analytical rigor, covering the definition of an ellipse with eccentricity \(e < 1\), derivation of standard forms \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (horizontal) and \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) (vertical), key elements including foci \((\pm c, 0)\), vertices \((\pm a, 0)\), co-vertices \((0, \pm b)\), directrices \(x = \pm \frac{a}{e}\), eccentricity \(e = \frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}\), length of latus rectum \(\frac{2b^2}{a}\), translated central forms \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), equation of tangent \(\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1\), tangency condition \(c^2 = a^2 m^2 + b^2\), and equations of normals.
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Exercise Overview

Exercise 6.4 of Unit 6: Conics in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum focuses on the ellipse, defined as the locus of a point moving in a plane such that the sum of its distances from two fixed points (foci (F) and (F')) remains constant ((|PF| + |PF'| = 2a)), corresponding to an eccentricity (e = rac{c}{a} < 1). The standard central equation of a horizontal ellipse is ( rac{x^2}{a^2} + rac{y^2}{b^2} = 1) (where (a > b > 0) and (c^2 = a^2 - b^2)), featuring centre at (C(0, 0)), foci at ((pm c, 0)), major axis of length (2a) along the (x)-axis with vertices ((pm a, 0)), minor axis of length (2b) along the (y)-axis with co-vertices ((0, pm b)), directrices (x = pm rac{a}{e}), and latus rectum length ( rac{2b^2}{a}). For a vertical ellipse ( rac{x^2}{b^2} + rac{y^2}{a^2} = 1), the major axis lies along the (y)-axis with foci ((0, pm c)) and directrices (y = pm rac{a}{e}).

The exercise establishes two key theorems for translated ellipses: Theorem 1 covers translated horizontal ellipses ( rac{(x - h)^2}{a^2} + rac{(y - k)^2}{b^2} = 1) having centre (C(h, k)), foci ((h pm c, k)), vertices ((h pm a, k)), and directrices (x = h pm rac{a}{e}); Theorem 2 covers translated vertical ellipses ( rac{(x - h)^2}{b^2} + rac{(y - k)^2}{a^2} = 1) with centre (C(h, k)), foci ((h, k pm c)), and directrices (y = k pm rac{a}{e}). Furthermore, students derive the equation of the tangent at (P(x_1, y_1)) as ( rac{xx_1}{a^2} + rac{yy_1}{b^2} = 1), the condition of tangency for (y = mx + c) as (c^2 = a^2 m^2 + b^2) producing the slope form (y = mx pm sqrt{a^2 m^2 + b^2}) with contact points (left(mp rac{a^2 m}{c}, pm rac{b^2}{c} ight)), and the equation of the normal line.

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 6.4 is a frequent source of Section B Short Questions (extracting foci, vertices, directrices, eccentricity, and latus rectum; constructing equations from given geometric constraints) and Section C Long Questions (completing squares for translated second-degree curves, calculating tangent/normal equations, and finding points of contact). These step-by-step solutions provide rigorous algebraic workflows, complete square completions, and exact coordinate derivations for every textbook question.

FAQs

Are these Exercise 6.4 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 all ellipse concepts and problems in Unit 6, Exercise 6.4 for the 2026-2027 session.
Do these solutions cover every question in Exercise 6.4?
Yes, every question in Exercise 6.4 is completely solved, including finding ellipse elements (foci, vertices, co-vertices, directrices, eccentricity, latus rectum), constructing ellipse equations from given conditions, translated center equations, and tangent lines.
Are the solutions step-by-step?
Yes. Each problem details the exact algebraic steps-identifying horizontal (a > b) vs vertical (b > a) orientation, applying the fundamental relation c² = a² - b² (or c² = b² - a²), calculating eccentricity e = c/a, setting up directrices, and applying condition c² = a²m² + b².
Can students read these solutions online?
Yes, the complete Exercise 6.4 solution set is available to read online directly on this page on MaryamNotes.pk with zero software download or registration required.
Are these solutions based on the latest PCTB / PECTAA syllabus?
Yes, they are strictly aligned with the updated PCTB and PECTAA Mathematics curriculum for Class 12 HSSC Part 2 for the 2026-2027 academic year.
Are these solutions free for students?
Yes, Exercise 6.4 solutions on this page are completely free for all Intermediate Part 2 students across Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, and all other Punjab boards.