Exercise 6.3 Class 12 Math Solutions - Unit 6 Conics, Punjab Board
Step-by-step solutions to Exercise 6.3, Unit 6: Conics, for Class 12 Punjab Board students. Covers the definition of a parabola with eccentricity e = 1, derivation of standard equation y² = 4ax, key elements (focus, vertex, directrix, axis of symmetry, latus rectum length 4a), the four standard orientations, and equations of tangents and normals including the tangency condition c = a/m, worked out question by question.
Exercise Overview
Exercise 6.3 of Unit 6: Conics in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum introduces the parabola as the conic section with eccentricity (e = 1), defined geometrically as the set of all points in a plane equidistant from a fixed point called the focus (F) and a fixed line called the directrix (L). The standard equation (y^2 = 4ax) ((a > 0)) represents a parabola opening towards the positive (x)-axis, having its vertex at the origin (V(0, 0)), focus at (F(a, 0)), directrix line (x + a = 0) (or (x = -a)), and axis of symmetry along the (x)-axis ((y = 0)). The focal chord perpendicular to the axis of symmetry is the latus rectum, having a total length of (4a), with parametric equations given by (x = at^2) and (y = 2at).
The exercise systematically explores the four standard orientations of a parabola: right-opening ((y^2 = 4ax)), left-opening ((y^2 = -4ax) with focus ((-a, 0)) and directrix (x = a)), upward-opening ((x^2 = 4ay) with focus ((0, a)) and directrix (y = -a)), and downward-opening ((x^2 = -4ay) with focus ((0, -a)) and directrix (y = a)). Furthermore, students determine equations of tangents and normals to a parabola: the tangent at (P(x_1, y_1)) is (yy_1 = 2a(x + x_1)), the condition of tangency for a line (y = mx + c) is (c = rac{a}{m}) yielding the slope-form tangent (y = mx + rac{a}{m}) with contact point (left( rac{a}{m^2}, rac{2a}{m} ight)), and the equation of the corresponding 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 6.3 is a high-yield source of Section B Short Questions (extracting focus, vertex, directrix, and latus rectum; formulating parabola equations from given geometric data) and Section C Long Questions (finding tangent and normal lines, evaluating intersections, and solving reflection/focal property problems). These step-by-step solutions provide rigorous algebraic calculations and clear geometric sketches for every textbook problem.