Exercise 6.2 Class 12 Math Solutions - Unit 6 Conics, Punjab Board
Step-by-step solutions to Exercise 6.2, Unit 6: Conics, for Class 12 Punjab Board students. Covers equations of tangents and normals to a circle, standard replacement rules, position of a point with respect to a circle, line-circle intersection cases, condition of tangency c² = r²(1 + m²), coordinates of point of contact, and formulas for the length of tangents from an external point, worked out question by question.
Exercise Overview
Exercise 6.2 of Unit 6: Conics in Class 12 Mathematics under the Punjab Board PCTB / PECTAA curriculum focuses on the geometric properties of tangents and normals to a circle. For a standard circle (x^2 + y^2 = r^2), the equation of the tangent at point (P(x_1, y_1)) is (xx_1 + yy_1 = r^2), while the equation of the normal is (y - y_1 = rac{y_1}{x_1}(x - x_1)), simplifying to (x y_1 - y x_1 = 0) (demonstrating that the normal to a circle always passes through its centre). For the general circle (x^2 + y^2 + 2gx + 2fy + c = 0), the equation of the tangent at (P(x_1, y_1)) is constructed via the fundamental replacement rules ((x^2 o xx_1), (y^2 o yy_1), (2x o x + x_1), (2y o y + y_1)) as (xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0), with its normal passing through ((-g, -f)) having slope ( rac{y_1 + f}{x_1 + g}).
The exercise thoroughly investigates the position of a point (P(x_1, y_1)) with respect to a circle using the power value (S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c): the point lies outside if (S_1 > 0), on the circle if (S_1 = 0), and inside if (S_1 < 0). The intersection of a line (y = mx + c) with a circle (x^2 + y^2 = r^2) is analyzed through the discriminant of the resulting quadratic: two real points ((Delta > 0)), one contact point / tangent line ((Delta = 0 implies c^2 = r^2(1 + m^2))), or no real intersection ((Delta < 0)). The coordinates of the point of contact are derived as (left(- rac{mr^2}{c}, rac{r^2}{c} ight)), and the length of a tangent drawn from an external point (P(x_1, y_1)) is given by (L = sqrt{S_1} = sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}).
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 6.2 is essential for Section B Short Questions (evaluating point position, finding tangent length, writing tangent/normal equations) and Section C Long Questions (finding tangents from an external point, finding common tangents, and applying tangency condition proofs). These step-by-step solutions provide rigorous derivations, quadratic discriminant checks, and complete algebraic details for every textbook question.