Exercise 6.6 Class 12 Math Solutions - Unit 6 Conics, Punjab Board
Step-by-step solutions to Exercise 6.6, Unit 6: Conics, for Class 12 Punjab Board students. Covers real-world applications of conic sections including quantum atomic physics (elliptical electron orbits in Sommerfeld's model), celestial mechanics, orbital parameters, and satellite orbits around the Earth (apogee, perigee, eccentricity, major axis), worked out question by question.
Exercise Overview
Exercise 6.6 concludes Unit 6: Conics in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum by connecting theoretical coordinate geometry to real-world physical and engineering phenomena. In atomic physics, the Bohr-Sommerfeld model extends circular atomic trajectories to quantized elliptical orbits with the atomic nucleus situated at one focal point. Students utilize the relationship between the semi-major axis (a), the semi-minor axis (b), the principal quantum number (n), and the azimuthal quantum number (k) to establish the ratio ( rac{b}{a} = rac{k}{n}) and construct exact Cartesian trajectory profiles for orbiting electrons.
In astrophysics and aerospace engineering, Kepler's First Law governs the orbital motion of artificial satellites, moons, and planets, demonstrating that celestial bodies traverse elliptical orbits with the central mass (such as Earth or the Sun) at one focus. The exercise details calculations for key orbital metrics: the distance of closest approach (perigee (r_p = a - c = a(1 - e))), the distance of farthest departure (apogee (r_a = a + c = a(1 + e))), the semi-major axis (a = rac{r_a + r_p}{2}), the focal distance (c = rac{r_a - r_p}{2}), the orbital eccentricity (e = rac{r_a - r_p}{r_a + r_p}), and the semi-minor axis (b = sqrt{a^2 - c^2} = sqrt{r_a r_p}).
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 6.6 provides critical SLO-aligned conceptual questions and applied Section C Long Questions. These step-by-step solutions provide clear physical interpretations, comprehensive algebraic transformations, and exact numerical answers for every textbook problem.