Exercise 6.1 Class 12 Math Solutions - Unit 6 Conics, Punjab Board

Exercise 6.1 opens Unit 6: Conics in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum by introducing the circle as the foundational conic section formed by intersecting a right circular cone with a plane perpendicular to the cone's axis. The exercise establishes the standard central equation of a circle ((x - h)^2 + (y - k)^2 = r^2) with centre (C(h, k)) and radius (r), as well as the standard parametric equations (x = h + rcos heta) and (y = k + rsin heta). Expanding the central form yields the general second-degree equation of a circle (x^2 + y^2 + 2gx + 2fy + c = 0), whose centre is (C(-g, -f)) and radius is (r = sqrt{g^2 + f^2 - c}). Students categorize circles as real ((g^2 + f^2 - c > 0)), point ((g^2 + f^2 - c = 0)), or imaginary ((g^2 + f^2 - c < 0)).

The exercise systematically derives the equation of a circle when the endpoints of a diameter (A(x_1, y_1)) and (B(x_2, y_2)) are given, expressed via vector orthogonality as ((x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0). Furthermore, students master the four classical geometric constraint cases required to determine the unique three parameters (g, f,) and (c): Case 1 (a circle passing through three non-collinear points), Case 2 (a circle passing through two points with its centre lying on a specified line), Case 3 (a circle passing through two points with a given tangent line at one of those points), and Case 4 (a circle passing through two points and touching a given straight 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 6.1 provides essential Section B Short Questions (extracting centre/radius, forming parametric representations, diameter equations) and high-yield Section C Long Questions (solving simultaneous three-parameter systems for circles under geometric constraints). These step-by-step solutions provide rigorous algebraic calculations and clear geometric reasoning for every textbook question.